1 00:00:00,000 --> 00:00:00,480 2 00:00:00,480 --> 00:00:04,290 Let's say we've got the function, f of x is equal 3 00:00:04,290 --> 00:00:08,780 to the natural log of x to the fourth plus 27. 4 00:00:08,779 --> 00:00:11,059 And all we want to do is take its first and second 5 00:00:11,060 --> 00:00:15,199 derivatives, and use as much of our techniques as we have at 6 00:00:15,199 --> 00:00:17,320 our disposal to attempt to graph it without a 7 00:00:17,320 --> 00:00:18,210 graphing calculator. 8 00:00:18,210 --> 00:00:20,560 If we have time, I'll take out the graphing calculator and 9 00:00:20,559 --> 00:00:22,809 see if our answer matches up. 10 00:00:22,809 --> 00:00:25,299 So a good place to start is to take the first 11 00:00:25,300 --> 00:00:26,420 derivative of this. 12 00:00:26,420 --> 00:00:28,040 So let me do that over here. 13 00:00:28,039 --> 00:00:31,669 So the derivative of f. 14 00:00:31,670 --> 00:00:34,870 Well, you take the derivative of the inside, so take the 15 00:00:34,869 --> 00:00:39,899 derivative of that right there, which is 4x to the third, and 16 00:00:39,899 --> 00:00:44,960 then multiply it times the derivative of the outside, 17 00:00:44,960 --> 00:00:46,399 with respect to the inside. 18 00:00:46,399 --> 00:00:48,939 So the derivative of the natural log of x is 1over x. 19 00:00:48,939 --> 00:00:51,579 So the derivative of this whole thing with respect to this 20 00:00:51,579 --> 00:00:56,039 inside expression is going to be, so times 1 over x 21 00:00:56,039 --> 00:00:58,219 to the fourth plus 27. 22 00:00:58,219 --> 00:01:01,659 If you found that confusing, you might want to rewatch 23 00:01:01,659 --> 00:01:03,459 the chain rule videos. 24 00:01:03,460 --> 00:01:05,290 But that's the first derivative of our function. 25 00:01:05,290 --> 00:01:10,670 I could rewrite this, this is equal to 4x to the third over 26 00:01:10,670 --> 00:01:12,850 x to the fourth, plus 27. 27 00:01:12,849 --> 00:01:17,399 Or I could write it as 4x to the third times x to the 28 00:01:17,400 --> 00:01:21,000 fourth, plus 27 to the negative 1. 29 00:01:21,000 --> 00:01:23,129 All three of these expressions are equivalent. 30 00:01:23,129 --> 00:01:25,949 I'm just the writing, I multiplied it out, or I could 31 00:01:25,950 --> 00:01:27,670 write this as a negative exponent, or I could write 32 00:01:27,670 --> 00:01:29,870 this as a fraction, with this in the denominator. 33 00:01:29,870 --> 00:01:30,740 They're all the equivalent. 34 00:01:30,739 --> 00:01:31,619 So that's our first derivative. 35 00:01:31,620 --> 00:01:35,060 Let's do our second derivative. 36 00:01:35,060 --> 00:01:36,990 Our second derivative, this looks like it'll get 37 00:01:36,989 --> 00:01:39,469 a little bit hairier. 38 00:01:39,469 --> 00:01:42,219 So our second derivative is the derivative of this. 39 00:01:42,219 --> 00:01:44,590 So it's equal to, we can now use the product rule, is the 40 00:01:44,590 --> 00:01:48,760 derivative of this first expression, times the 41 00:01:48,760 --> 00:01:49,590 second expression. 42 00:01:49,590 --> 00:01:53,260 So the derivative of this first expression, 3 times 4is 12. 43 00:01:53,260 --> 00:01:58,450 12x squared, right, we just decrement the 3 by 1, times the 44 00:01:58,450 --> 00:02:06,280 second expression, times x to the fourth plus 27 to the minus 45 00:02:06,280 --> 00:02:12,379 1, and then to that, we want to add just the first expression, 46 00:02:12,379 --> 00:02:15,900 not its derivative, so just 4x to the third, times 47 00:02:15,900 --> 00:02:17,305 the derivative of the second expression. 48 00:02:17,305 --> 00:02:19,379 And the derivative of the second expression, we could 49 00:02:19,379 --> 00:02:22,939 take the derivative of the inside, which is just 4x to the 50 00:02:22,939 --> 00:02:28,689 third, the derivative of 27 is just 0, so times 4x to the 51 00:02:28,689 --> 00:02:32,669 third, times the derivative of this whole thing with 52 00:02:32,669 --> 00:02:33,789 respect to the inside. 53 00:02:33,789 --> 00:02:37,650 So times, so you take this exponent, put out front, so 54 00:02:37,650 --> 00:02:44,289 times minus 1, times this whole thing, x to the fourth plus 55 00:02:44,289 --> 00:02:49,929 27 to the, we decrement this by one more, so minus two. 56 00:02:49,930 --> 00:02:53,420 So let's see if I can simplify this expression a little bit. 57 00:02:53,419 --> 00:03:00,589 So this is equal to, so this right here is equal to 12x 58 00:03:00,590 --> 00:03:07,400 squared over this thing, x to the fourth plus 27, and then, 59 00:03:07,400 --> 00:03:09,219 let's see, if we multiply we're going to have a minus here, so 60 00:03:09,219 --> 00:03:14,400 it's a minus, you multiply these two guys, 4 times 4is 16, 61 00:03:14,400 --> 00:03:18,659 16x to the third times x to the third is x to the sixth, 62 00:03:18,659 --> 00:03:20,379 over this thing squared. 63 00:03:20,379 --> 00:03:26,620 Over x to the fourth plus 27 squared. 64 00:03:26,620 --> 00:03:28,599 That's just another way to rewrite that expression 65 00:03:28,599 --> 00:03:29,379 right there, right? 66 00:03:29,379 --> 00:03:31,810 To the minus 2, you just put in the denominator and make it 67 00:03:31,810 --> 00:03:34,150 into a positive 2 in the denominator. 68 00:03:34,150 --> 00:03:35,610 Same thing. 69 00:03:35,610 --> 00:03:37,860 Now, if you've seen these problems in the past, we 70 00:03:37,860 --> 00:03:39,480 always want to set these things equal to 0. 71 00:03:39,479 --> 00:03:41,090 We want to solve for x equals 0. 72 00:03:41,090 --> 00:03:45,750 So it'll be useful to have this expressed as just one fraction, 73 00:03:45,750 --> 00:03:48,460 instead of the difference or the sum of two fractions. 74 00:03:48,460 --> 00:03:50,879 So what we can do, is we could have a common denominator. 75 00:03:50,879 --> 00:03:54,169 So we could multiply both the numerator and denominator of 76 00:03:54,169 --> 00:03:56,829 this expression by x to the fourth plus 27, and 77 00:03:56,830 --> 00:03:57,290 what do we get? 78 00:03:57,289 --> 00:04:00,829 So this is equal to, so if we multiply this first expression, 79 00:04:00,830 --> 00:04:06,730 times x to the fourth plus 27, we get 12x squared, times 80 00:04:06,729 --> 00:04:09,289 x to the fourth, plus 27. 81 00:04:09,289 --> 00:04:12,340 And then in the denominator, you have x to the 82 00:04:12,340 --> 00:04:15,830 fourth plus 27 squared. 83 00:04:15,830 --> 00:04:18,220 All I did, I multiplied this numerator and this denominator 84 00:04:18,220 --> 00:04:19,610 by x to the the fourth plus 27. 85 00:04:19,610 --> 00:04:20,610 I didn't change it. 86 00:04:20,610 --> 00:04:22,600 And then we have that second term. 87 00:04:22,600 --> 00:04:30,310 Minus 16 x to the sixth over x to the fourth plus 27 squared. 88 00:04:30,310 --> 00:04:31,019 The whole reason why did that? 89 00:04:31,019 --> 00:04:32,740 Now I have a common denominator, now I can 90 00:04:32,740 --> 00:04:34,250 just add the numerators. 91 00:04:34,250 --> 00:04:40,230 So this is going to be equal to, let's see. 92 00:04:40,230 --> 00:04:44,360 The denominator, we know what the denominator is, it is x to 93 00:04:44,360 --> 00:04:48,410 the fourth plus 27 squared. 94 00:04:48,410 --> 00:04:49,330 That's our denominator. 95 00:04:49,329 --> 00:04:50,479 And then we can multiply this out. 96 00:04:50,480 --> 00:04:52,580 This is 12x squared times x to the fourth. 97 00:04:52,579 --> 00:04:59,039 That's 12x to the sixth, plus 27 times 12. 98 00:04:59,040 --> 00:05:01,150 I don't even feel like multiplying 27 times 12,so 99 00:05:01,149 --> 00:05:02,060 I'll just write that out. 100 00:05:02,060 --> 00:05:09,300 So plus 27 times 12x squared, I just multiplied the 12x squared 101 00:05:09,300 --> 00:05:14,569 times the 27, and then minus 16x to the sixth minus 102 00:05:14,569 --> 00:05:17,430 16x to the sixth. 103 00:05:17,430 --> 00:05:20,000 And this simplifies to, let's see if I can simplify 104 00:05:20,000 --> 00:05:21,240 this even further. 105 00:05:21,240 --> 00:05:24,050 7x to the sixth here, x to the sixth here. 106 00:05:24,050 --> 00:05:27,720 So this is equal to, do this in pink. 107 00:05:27,720 --> 00:05:31,220 This is equal to the 27 times 12x squared, I don't feel like 108 00:05:31,220 --> 00:05:35,970 figuring that out right now, times 12x squared, and then you 109 00:05:35,970 --> 00:05:39,140 have minus 16x to the sixth and plus 12x to the sixth. 110 00:05:39,139 --> 00:05:41,599 So you add those two, you get minus 4. 111 00:05:41,600 --> 00:05:48,900 12 minus 15is minus 4, x to the sixth, all of that over x to 112 00:05:48,899 --> 00:05:57,159 the fourth plus 27 plus 27 squared. 113 00:05:57,160 --> 00:05:59,000 And that is our second derivative. 114 00:05:59,000 --> 00:06:02,189 Now, we've done all of the derivatives, and this was 115 00:06:02,189 --> 00:06:03,800 actually a pretty hairy problem. 116 00:06:03,800 --> 00:06:06,860 And now we can solve for when the first and the second 117 00:06:06,860 --> 00:06:10,305 derivatives equal 0, and we'll have our candidate, well, we'll 118 00:06:10,305 --> 00:06:12,810 know our critical points, and then we'll have our candidate 119 00:06:12,810 --> 00:06:14,780 inflection points, and see if we can make any 120 00:06:14,779 --> 00:06:15,629 headway from there. 121 00:06:15,629 --> 00:06:18,939 So first, let's see where our first derivative is equal to 0, 122 00:06:18,939 --> 00:06:20,240 and get our critical points. 123 00:06:20,240 --> 00:06:23,680 Or at least maybe, also maybe, where it's undefined. 124 00:06:23,680 --> 00:06:25,329 So this is equal to 0. 125 00:06:25,329 --> 00:06:28,519 If we want to set, if the only place that this can equal to 0 126 00:06:28,519 --> 00:06:30,740 is if this numerator is equal to 0. 127 00:06:30,740 --> 00:06:34,490 This denominator, actually, if we are assuming we're dealing 128 00:06:34,490 --> 00:06:37,490 with real numbers, this term right here is always going to 129 00:06:37,490 --> 00:06:41,129 be greater than or equal to 0 for any value of x, because 130 00:06:41,129 --> 00:06:42,490 it's an even exponent. 131 00:06:42,490 --> 00:06:44,790 So this thing can never equals 0, right, because you're 132 00:06:44,790 --> 00:06:47,410 adding 27 to something that's non-negative. 133 00:06:47,410 --> 00:06:49,540 So this will never equal 0, so this will also 134 00:06:49,540 --> 00:06:50,850 never be undefined. 135 00:06:50,850 --> 00:06:54,890 So there's no undefined critical points here, but 136 00:06:54,889 --> 00:06:58,550 we could set the numerator equal to 0 pretty easily. 137 00:06:58,550 --> 00:07:02,060 If we wanted to set this equal to 0, we just say 4x to the 138 00:07:02,060 --> 00:07:04,899 third is equal to 0, and we know what x-value will make 139 00:07:04,899 --> 00:07:08,279 that equal to 0, x has to be equal to 0. 140 00:07:08,279 --> 00:07:10,229 4 times something to the third is equal to 0, that 141 00:07:10,230 --> 00:07:11,759 something has to be 0. 142 00:07:11,759 --> 00:07:14,769 x to the third has to be 0, x has to be 0. 143 00:07:14,769 --> 00:07:19,589 So we can write, f prime of 0 is equal to 0. 144 00:07:19,589 --> 00:07:22,349 So 0 is a critical point. 145 00:07:22,350 --> 00:07:23,970 0 is a critical point. 146 00:07:23,970 --> 00:07:26,940 147 00:07:26,939 --> 00:07:29,189 The slope at 0 is 0. 148 00:07:29,189 --> 00:07:32,060 We don't know if it's a maximum or a minimum, or 149 00:07:32,060 --> 00:07:33,379 an inflection point yet. 150 00:07:33,379 --> 00:07:36,529 151 00:07:36,529 --> 00:07:37,784 We'll explore it a little bit more. 152 00:07:37,785 --> 00:07:39,310 And actually, just so we get the coordinate, 153 00:07:39,310 --> 00:07:40,379 what's the coordinate? 154 00:07:40,379 --> 00:07:45,149 The coordinate x is 0, and then y is the natural log-- if x is 155 00:07:45,149 --> 00:07:49,620 0, this just turns out, it's the a natural log of 27. 156 00:07:49,620 --> 00:07:51,879 Let me figure out what that is, I'll get the calculator out. 157 00:07:51,879 --> 00:07:53,740 I said I wouldn't use a graphing calculator, but I can 158 00:07:53,740 --> 00:07:54,910 use a regular calculator. 159 00:07:54,910 --> 00:07:59,870 So27, if I were to take the natural log of that, for 160 00:07:59,870 --> 00:08:01,420 our purposes let's just call it 3.3. 161 00:08:01,420 --> 00:08:03,759 We're just trying to get the general shape of the graph. 162 00:08:03,759 --> 00:08:04,189 So 3.3. 163 00:08:04,189 --> 00:08:07,709 164 00:08:07,709 --> 00:08:11,049 Well, we could just say 2.9 and it kept going. 165 00:08:11,050 --> 00:08:13,020 So this is a critical point right here. 166 00:08:13,019 --> 00:08:15,209 The slope is 0 here. 167 00:08:15,209 --> 00:08:20,259 Slope is equal to 0 at x is equal to 0. 168 00:08:20,259 --> 00:08:22,110 So this is one thing we want to block off. 169 00:08:22,110 --> 00:08:23,930 And let's see if we can find any candidate 170 00:08:23,930 --> 00:08:25,170 inflection points. 171 00:08:25,170 --> 00:08:27,910 And remember, candidate inflection points are where the 172 00:08:27,910 --> 00:08:29,510 second derivative equals 0. 173 00:08:29,509 --> 00:08:32,039 Now if the second derivative equals 0, that doesn't tell 174 00:08:32,039 --> 00:08:33,789 us that those are definitely inflection points. 175 00:08:33,789 --> 00:08:35,909 Let me make this very clear. 176 00:08:35,909 --> 00:08:39,519 If, let me do it in a new color. 177 00:08:39,519 --> 00:08:52,159 If x is inflection, then the second derivative at x is 178 00:08:52,159 --> 00:08:54,089 going to be equal to 0. 179 00:08:54,090 --> 00:08:58,910 Because you're having a change concavity. 180 00:08:58,909 --> 00:09:02,829 You have a change in the slope, goes from either increasing 181 00:09:02,830 --> 00:09:05,340 to decreasing or from decreasing to increasing. 182 00:09:05,340 --> 00:09:09,350 But if the derivative is equal to 0, the second derivative is 183 00:09:09,350 --> 00:09:11,899 equal to 0, you cannot assume that is an inflection point. 184 00:09:11,899 --> 00:09:14,389 So what we're going to do is, we're going to find all of the 185 00:09:14,389 --> 00:09:19,139 point at which this is true, and then see if we actually do 186 00:09:19,139 --> 00:09:22,230 have a sign change in the second derivative of that 187 00:09:22,230 --> 00:09:25,279 point, and only if you have a sign change, then you can say 188 00:09:25,279 --> 00:09:26,549 it's an inflection point. 189 00:09:26,549 --> 00:09:28,169 So let's see if we can do that. 190 00:09:28,169 --> 00:09:30,219 So just because a second derivative is 0, that by 191 00:09:30,220 --> 00:09:32,550 itself does not tell you it's an inflection point. 192 00:09:32,549 --> 00:09:36,209 It has to have a second derivative of 0, and when you 193 00:09:36,210 --> 00:09:40,190 go above or below that x, the second derivative has to 194 00:09:40,190 --> 00:09:41,500 actually change signs. 195 00:09:41,500 --> 00:09:42,139 Only then. 196 00:09:42,139 --> 00:09:54,169 So we can say, if f prime changes signs around x, 197 00:09:54,169 --> 00:09:56,889 then we can say that x is an inflection. 198 00:09:56,889 --> 00:10:00,225 199 00:10:00,225 --> 00:10:03,070 And if it's changing signs around x, then it's definitely 200 00:10:03,070 --> 00:10:06,470 going to be 0 right at x, but you have to actually see that 201 00:10:06,470 --> 00:10:09,350 if it's negative before x, has to be positive after x,or if 202 00:10:09,350 --> 00:10:11,736 it's positive before x, has to be negative after x. 203 00:10:11,736 --> 00:10:13,269 So let's test that out. 204 00:10:13,269 --> 00:10:15,610 So the first thing we need to do is find 205 00:10:15,610 --> 00:10:16,490 these candidate points. 206 00:10:16,490 --> 00:10:19,430 Remember, the candidate points are where the second 207 00:10:19,429 --> 00:10:21,159 derivative is equal to 0. 208 00:10:21,159 --> 00:10:23,569 We're going to find those points, and then see if this 209 00:10:23,570 --> 00:10:25,670 is true, that the sign actually changes. 210 00:10:25,669 --> 00:10:29,750 We want to find where this thing over here is equal to 0. 211 00:10:29,750 --> 00:10:32,450 And once again, for this to be equal to 0, the numerator 212 00:10:32,450 --> 00:10:33,390 has to be equal to 0. 213 00:10:33,389 --> 00:10:36,199 This denominator can never be equal to 0 if we're dealing 214 00:10:36,200 --> 00:10:38,570 with real numbers, which I think is a fair assumption. 215 00:10:38,570 --> 00:10:43,450 So let's see where this our numerator can be equal to 0 216 00:10:43,450 --> 00:10:45,650 for the second derivative. 217 00:10:45,649 --> 00:10:48,610 So let's set the numerator of the second derivative. 218 00:10:48,610 --> 00:10:54,899 27 times 12x squared minus 4x to the sixth is equal to 0. 219 00:10:54,899 --> 00:10:56,209 Remember, that's just the numerator of our 220 00:10:56,210 --> 00:10:57,250 second derivative. 221 00:10:57,250 --> 00:10:59,360 Any x that makes the numerator 0 is making 222 00:10:59,360 --> 00:11:01,580 the second derivative 0. 223 00:11:01,580 --> 00:11:06,500 So let's factor out a 4x squared. 224 00:11:06,500 --> 00:11:09,059 So 4x squared. 225 00:11:09,059 --> 00:11:14,139 Now we'll have 27 times, if we factor 4 out of the 12, we'll 226 00:11:14,139 --> 00:11:17,899 just get a 3, and we factored out the x squared, minus, we 227 00:11:17,899 --> 00:11:21,120 factored out the 4, we factored out an x squared, so we have x 228 00:11:21,120 --> 00:11:24,580 to the fourth is equal to 0. 229 00:11:24,580 --> 00:11:29,480 So the x's that will make this equal to 0 will satisfy either, 230 00:11:29,480 --> 00:11:35,180 I'll switch colors, either 4x squared is equal to 0, or, now 231 00:11:35,179 --> 00:11:37,059 27 times 3, I can do that in my head. 232 00:11:37,059 --> 00:11:38,169 That's 81. 233 00:11:38,169 --> 00:11:43,059 20 times 3 is 60, 7 times 3 is 21, 60 plus 21is 81. 234 00:11:43,059 --> 00:11:48,919 Or 81 minus x to the fourth is equal to 0. 235 00:11:48,919 --> 00:11:51,409 Any x that satisfies either of these will make this 236 00:11:51,409 --> 00:11:52,699 entire expression equal 0. 237 00:11:52,700 --> 00:11:54,180 Because if this thing is 0, the whole thing is 238 00:11:54,179 --> 00:11:54,754 going to be equal to 0. 239 00:11:54,754 --> 00:11:56,750 If this thing is 0, the whole thing is going 240 00:11:56,750 --> 00:11:57,445 to be equal to 0. 241 00:11:57,445 --> 00:11:59,640 Let me be clear, this is 81 right there. 242 00:11:59,639 --> 00:12:00,759 So let's solve this. 243 00:12:00,759 --> 00:12:07,460 This is going to be 0 when x is equal to 0, itself. 244 00:12:07,460 --> 00:12:10,560 This is going to be equal to 0 when x, let's see. 245 00:12:10,559 --> 00:12:12,589 If we add x to the fourth to both sides, you get x to 246 00:12:12,590 --> 00:12:15,060 the fourth is equal to 81. 247 00:12:15,059 --> 00:12:18,069 If we take the square root of both sides of this, you get x 248 00:12:18,070 --> 00:12:23,090 squared is equal to 9, or so you get x is plus or minus 3. 249 00:12:23,090 --> 00:12:29,590 x is equal to plus or minus three. 250 00:12:29,590 --> 00:12:32,639 So these are our candidate inflection points, x is equal 251 00:12:32,639 --> 00:12:36,769 to 0, x is equal to plus 3, or x is equal to minus 3. 252 00:12:36,769 --> 00:12:39,149 So what we have to do now, is to see whether the second 253 00:12:39,149 --> 00:12:42,360 derivative changes signs around these points in order to be 254 00:12:42,360 --> 00:12:44,710 able to label them inflection points. 255 00:12:44,710 --> 00:12:49,129 So what happens when x is slightly below 0? 256 00:12:49,129 --> 00:12:52,289 So let's take the situation, let's do all the scenarios. 257 00:12:52,289 --> 00:12:54,719 What happens when x is slightly below 0? 258 00:12:54,720 --> 00:12:58,620 Not all of them, necessarily, but if x is like 0.1. 259 00:12:58,620 --> 00:13:01,490 What is the second derivative going to be doing? 260 00:13:01,490 --> 00:13:07,230 If x is 0.1, or if x is minus 0.1, this term right here is 261 00:13:07,230 --> 00:13:12,080 going to be positive, and then this is going to be 81 262 00:13:12,080 --> 00:13:13,990 minus 0.1 to the fourth. 263 00:13:13,990 --> 00:13:16,889 So that's going to be a very small number, right? 264 00:13:16,889 --> 00:13:19,519 So it's going to be some positive number times 81 265 00:13:19,519 --> 00:13:20,529 minus a small number. 266 00:13:20,529 --> 00:13:22,299 So it's going to be a positive number. 267 00:13:22,299 --> 00:13:26,929 So when x is less than 0, or just slightly less than 0, our 268 00:13:26,929 --> 00:13:31,739 second derivative is positive. 269 00:13:31,740 --> 00:13:34,519 Now what happens when x is slightly larger? 270 00:13:34,519 --> 00:13:36,319 When I write this notation, I want to be careful, I mean, 271 00:13:36,320 --> 00:13:39,960 really, just right below 0. 272 00:13:39,960 --> 00:13:43,030 Now when x is right above 0, what happens? 273 00:13:43,029 --> 00:13:47,039 Let's say x was 0.01, or 0.1, positive 0.1. 274 00:13:47,039 --> 00:13:48,179 Well, it's going to be the same thing. 275 00:13:48,179 --> 00:13:49,829 Because in both cases, we're squaring, and 276 00:13:49,830 --> 00:13:50,759 we're taking the fourth. 277 00:13:50,759 --> 00:13:53,850 So you're kind of losing your sign information. 278 00:13:53,850 --> 00:13:57,290 So if x is 0.1, this thing is going to be a small 279 00:13:57,289 --> 00:13:58,469 positive number. 280 00:13:58,470 --> 00:14:00,850 You're going to be subtracting a very small positive number 281 00:14:00,850 --> 00:14:05,060 from 81, but 81 minus a small number is still going 282 00:14:05,059 --> 00:14:05,919 to be positive. 283 00:14:05,919 --> 00:14:09,809 So you're going to positive times a positive, so your 284 00:14:09,809 --> 00:14:13,529 second derivative is still going to be greater than 0. 285 00:14:13,529 --> 00:14:15,429 So something interesting here. 286 00:14:15,429 --> 00:14:20,939 f at your second derivative is 0 when x is equal to 0, but it 287 00:14:20,940 --> 00:14:22,980 is a not an inflection point. 288 00:14:22,980 --> 00:14:29,230 Because notice, the concavity did not change around 0. 289 00:14:29,230 --> 00:14:32,810 Our second derivative is positive as we approach 0 from 290 00:14:32,809 --> 00:14:36,289 the left, and it's positive as we approach 0 from the right. 291 00:14:36,289 --> 00:14:41,339 So in general, at 0, we're always, as we're near 0 from 292 00:14:41,340 --> 00:14:44,040 either direction, we're going to be concave upwards. 293 00:14:44,039 --> 00:14:48,819 So the fact that 0 is a critical point, and that we're 294 00:14:48,820 --> 00:14:52,600 always concave upward, as we approach 0 from either side, 295 00:14:52,600 --> 00:14:55,690 this tells us that this is a minimum point. 296 00:14:55,690 --> 00:15:00,760 297 00:15:00,759 --> 00:15:04,600 Because we're concave upwards all around 0. 298 00:15:04,600 --> 00:15:07,159 So 0 is not an inflection point. 299 00:15:07,159 --> 00:15:09,490 Let's see if positive and negative 3 are 300 00:15:09,490 --> 00:15:11,269 inflection points. 301 00:15:11,269 --> 00:15:14,519 And if you study this equation, let me write 302 00:15:14,519 --> 00:15:17,480 our-- and actually, I just want to be clear. 303 00:15:17,480 --> 00:15:19,050 I've just been using the numerator of the 304 00:15:19,049 --> 00:15:20,269 second derivative. 305 00:15:20,269 --> 00:15:23,879 The whole second derivative is this thing right here, but I've 306 00:15:23,879 --> 00:15:25,799 been ignoring the denominator because the denominator 307 00:15:25,799 --> 00:15:27,339 is always positive. 308 00:15:27,340 --> 00:15:29,139 So if we're trying to understand whether things are 309 00:15:29,139 --> 00:15:30,980 positive or negative, we just really have to determine 310 00:15:30,980 --> 00:15:33,000 whether the numerator is positive or negative. 311 00:15:33,000 --> 00:15:35,429 Because this expression right there is always positive. 312 00:15:35,429 --> 00:15:38,929 It's something to the second power. 313 00:15:38,929 --> 00:15:43,199 So let's test whether we have a change in concavity around x is 314 00:15:43,200 --> 00:15:45,610 equal to positive or negative 3. 315 00:15:45,610 --> 00:15:48,450 So remember, the numerator of our, let me just rewrite our 316 00:15:48,450 --> 00:15:50,850 second derivative, just so you see it here. f 317 00:15:50,850 --> 00:15:52,180 prime prime of x. 318 00:15:52,179 --> 00:15:54,169 The numerator is this thing right here. 319 00:15:54,169 --> 00:16:00,329 It's 4x squared times 81 minus x to the fourth. 320 00:16:00,330 --> 00:16:03,050 and the denominator was up here, x to the 321 00:16:03,049 --> 00:16:04,429 fourth plus 27 squared. 322 00:16:04,429 --> 00:16:10,539 323 00:16:10,539 --> 00:16:12,120 That was our second derivative. 324 00:16:12,120 --> 00:16:16,269 Let's see if this changes signs around positive or negative 3. 325 00:16:16,269 --> 00:16:19,939 And actually, we should get the same answer, because regardless 326 00:16:19,940 --> 00:16:22,440 of whether we put positive or negative 3 here. 327 00:16:22,440 --> 00:16:24,360 you lose all your sign information because you're 328 00:16:24,360 --> 00:16:25,835 taking it to the fourth power, you're taking it 329 00:16:25,835 --> 00:16:26,629 to the second power. 330 00:16:26,629 --> 00:16:28,850 And obviously, anything to the fourth power is always going to 331 00:16:28,850 --> 00:16:30,990 be positive, anything to the second power is always 332 00:16:30,990 --> 00:16:31,710 going to be negative. 333 00:16:31,710 --> 00:16:35,170 So when we do our test, if it's true for positive 3, it's 334 00:16:35,169 --> 00:16:37,949 probably going to be true for negative 3 as well. 335 00:16:37,950 --> 00:16:39,540 But let's just try it out. 336 00:16:39,539 --> 00:16:44,959 So when x is just a little bit less than positive 3, what's 337 00:16:44,960 --> 00:16:46,820 the sign of f prime prime of x? 338 00:16:46,820 --> 00:16:51,970 So it's going to be 4 times 9, or it's going to be 4 339 00:16:51,970 --> 00:16:53,879 times a positive number. 340 00:16:53,879 --> 00:16:55,929 It might be like 2.999, but this is still 341 00:16:55,929 --> 00:16:57,419 going to be positive. 342 00:16:57,419 --> 00:17:01,539 So this is going to be positive when x is approaching 3, and 343 00:17:01,539 --> 00:17:05,359 then this is going to be, well, if x is 3, this is 0, so x is 344 00:17:05,359 --> 00:17:07,419 a little bit less than 3. 345 00:17:07,420 --> 00:17:11,390 If x is a little bit less than 3, if it's like 2.9999, this 346 00:17:11,390 --> 00:17:14,120 number is going to be less than 81, so this is also 347 00:17:14,119 --> 00:17:15,459 going to be positive. 348 00:17:15,460 --> 00:17:17,860 And of course, the denominator is always positive. 349 00:17:17,859 --> 00:17:21,490 So as x is less than 3, is approaching from the left, 350 00:17:21,490 --> 00:17:22,759 we are concave upwards. 351 00:17:22,759 --> 00:17:24,650 This thing's going to be a positive. 352 00:17:24,650 --> 00:17:27,610 Then f prime prime is greater than 0. 353 00:17:27,609 --> 00:17:31,379 We are upwards, concave upwards. 354 00:17:31,380 --> 00:17:35,230 When x is just larger than 3, what's going to happen? 355 00:17:35,230 --> 00:17:37,819 Well, this first term is still going to be positive. 356 00:17:37,819 --> 00:17:41,220 But if x is just larger than 3, x to the fourth is going to be 357 00:17:41,220 --> 00:17:44,720 just larger than 81, and so this second term is going to be 358 00:17:44,720 --> 00:17:46,130 negative in that situation. 359 00:17:46,130 --> 00:17:48,676 360 00:17:48,675 --> 00:17:51,149 Let me do it ina new color. 361 00:17:51,150 --> 00:17:53,910 It's going to be negative when x is larger than 3. 362 00:17:53,910 --> 00:17:56,360 Because this is going to be larger than 81. 363 00:17:56,359 --> 00:17:59,229 So if this is negative and this is positive, then the whole 364 00:17:59,230 --> 00:18:00,980 thing is going to be negative, because this denominator is 365 00:18:00,980 --> 00:18:02,440 still going to be positive. 366 00:18:02,440 --> 00:18:05,250 So then f prime prime is going to be less than 0, so we're 367 00:18:05,250 --> 00:18:09,289 going to be concave downwards. 368 00:18:09,289 --> 00:18:10,700 One last one. 369 00:18:10,700 --> 00:18:16,279 What happens when x is just a greater than minus 3? 370 00:18:16,279 --> 00:18:19,470 So just being greater than minus 3, that's 371 00:18:19,470 --> 00:18:22,940 like minus 2.99999. 372 00:18:22,940 --> 00:18:25,279 So when you take minus 2.99 square it, you're going to get 373 00:18:25,279 --> 00:18:27,759 a positive number, so this is going to be positive. 374 00:18:27,759 --> 00:18:31,009 And if you take minus 2.99 to the fourth, that's going to be 375 00:18:31,009 --> 00:18:32,640 a little bit less than 81, right? 376 00:18:32,640 --> 00:18:35,130 Because 2.99 to the fourth is a little bit less than 81, so 377 00:18:35,130 --> 00:18:37,030 this is still going to be positive. 378 00:18:37,029 --> 00:18:39,920 So you have a positive times a positive divided by a positive, 379 00:18:39,920 --> 00:18:43,120 so you're going to be concave upwards, because your second 380 00:18:43,119 --> 00:18:45,089 derivative is going to be greater than 0. 381 00:18:45,089 --> 00:18:47,740 Concave upwards. 382 00:18:47,740 --> 00:18:52,000 And then finally, when x is just, just less than negative 383 00:18:52,000 --> 00:18:55,099 3, remember, when I write this down, I don't mean for all x's 384 00:18:55,099 --> 00:18:58,769 larger than negative 3,or all x's smaller than negative 3. 385 00:18:58,769 --> 00:19:01,129 There's actually no, well, I can't think of the notation 386 00:19:01,130 --> 00:19:04,910 that would say just, as we just approach three in this case, 387 00:19:04,910 --> 00:19:09,120 from the left, but what happens if we just go to minus 3.11? 388 00:19:09,119 --> 00:19:12,569 Or 3.01, I guess is a better one, or 3.1? 389 00:19:12,569 --> 00:19:15,039 Well, this term right here is going to be positive. 390 00:19:15,039 --> 00:19:19,319 But if we take minus 3.1 to the fourth, that's going to be 391 00:19:19,319 --> 00:19:21,759 larger than positive 81, right? 392 00:19:21,759 --> 00:19:24,079 The sign will become positive, it'll be larger than 81, so 393 00:19:24,079 --> 00:19:26,210 this'll become negative. 394 00:19:26,210 --> 00:19:28,539 So in that case as well, we'll have a positive times a 395 00:19:28,539 --> 00:19:32,409 negative divided by a positive, so then our second derivative 396 00:19:32,410 --> 00:19:34,100 is going to be negative. 397 00:19:34,099 --> 00:19:35,829 And so we're going to be downwards. 398 00:19:35,829 --> 00:19:38,339 399 00:19:38,339 --> 00:19:42,059 So I think we're ready to plot. so first of all, is x plus or 400 00:19:42,059 --> 00:19:43,119 minus 3 inflection points? 401 00:19:43,119 --> 00:19:43,869 Sure! 402 00:19:43,869 --> 00:19:47,799 As we approach x is equal to 3 from the left, we are concave 403 00:19:47,799 --> 00:19:52,809 upwards, and then as we cross 3, the second derivative is 0. 404 00:19:52,809 --> 00:19:54,549 The second derivative's 0, I lost it up here. 405 00:19:54,549 --> 00:19:57,879 The second derivative is 0. 406 00:19:57,880 --> 00:20:00,590 And then, as we go to the right of 3, we become 407 00:20:00,589 --> 00:20:01,399 concave downwards. 408 00:20:01,400 --> 00:20:04,050 So we got our sign change in the second derivative. 409 00:20:04,049 --> 00:20:06,000 So x is equal to 3. 410 00:20:06,000 --> 00:20:08,609 So 3 is definitely an inflection point, and the 411 00:20:08,609 --> 00:20:10,389 same argument could be made for negative 3. 412 00:20:10,390 --> 00:20:12,770 We switch signs as we cross 3. 413 00:20:12,769 --> 00:20:17,619 So these definitely are inflection points. 414 00:20:17,619 --> 00:20:19,919 Just so we get the exact coordinates, let's figure out 415 00:20:19,920 --> 00:20:24,050 what f of 3 is, or f of positive and negative 3. 416 00:20:24,049 --> 00:20:25,839 And then we're ready to graph. 417 00:20:25,839 --> 00:20:30,109 So first of all, we know that f, we know that the point 0, 418 00:20:30,109 --> 00:20:34,879 3.29, that this was a minimum. 419 00:20:34,880 --> 00:20:37,870 Because 0 was a critical point, the slope is 0 there, 420 00:20:37,869 --> 00:20:41,000 and because it's concave upwards all around 0. 421 00:20:41,000 --> 00:20:43,230 So 0 is definitely not an inflection point. 422 00:20:43,230 --> 00:20:46,970 And then we know that the points positive 3 and minus 3 423 00:20:46,970 --> 00:20:50,210 are inflection points, and in order to figure out their 424 00:20:50,210 --> 00:20:53,110 y-coordinates, we can just evaluate them. 425 00:20:53,109 --> 00:20:54,949 So they're actually going to have the same y-coordinates, 426 00:20:54,950 --> 00:20:57,680 because if you put a minus3 or positive 3 and take it to the 427 00:20:57,680 --> 00:20:59,009 fourth power, you're going to get the same thing. 428 00:20:59,009 --> 00:21:01,089 Let's figure out what they are. 429 00:21:01,089 --> 00:21:04,769 So if we take 3 to the fourth power, that's what, 81. 430 00:21:04,769 --> 00:21:09,990 81 plus 27 is equal to 108, and then we want to take 431 00:21:09,990 --> 00:21:11,160 the natural log of it. 432 00:21:11,160 --> 00:21:14,730 433 00:21:14,730 --> 00:21:17,130 Let's just say 4.7, just to get a rough idea. 434 00:21:17,130 --> 00:21:18,780 That's4.7. 435 00:21:18,779 --> 00:21:20,920 And that's true of whether we do positive or negative 3, 436 00:21:20,920 --> 00:21:22,300 because we took to the fourth power. 437 00:21:22,299 --> 00:21:27,720 So it's 4.7, 4.7. 438 00:21:27,720 --> 00:21:31,240 These are both inflection points. 439 00:21:31,240 --> 00:21:33,569 And we should be ready to graph it! 440 00:21:33,569 --> 00:21:35,250 Let's graph it. 441 00:21:35,250 --> 00:21:35,789 All right. 442 00:21:35,789 --> 00:21:42,089 Let me draw my axis, just like that. 443 00:21:42,089 --> 00:21:46,839 And this is my y-axis, this is my x-axis, this is y. 444 00:21:46,839 --> 00:21:49,740 You can even call it the f of x axis, if you like. 445 00:21:49,740 --> 00:21:50,819 This is x. 446 00:21:50,819 --> 00:21:53,750 And so the point 0, 3.29. 447 00:21:53,750 --> 00:22:01,210 Let's say this is 1, 2, 3, 4, 5, to the point 0, 3.29. 448 00:22:01,210 --> 00:22:05,289 That's 0, 1, 2, 3, a little bit above 3, it's right there. 449 00:22:05,289 --> 00:22:06,950 That's the minimum point. 450 00:22:06,950 --> 00:22:08,549 And then we're concave. 451 00:22:08,549 --> 00:22:12,230 The slope is 0 right there, we figured that out, because the 452 00:22:12,230 --> 00:22:13,360 first derivative was 0 there. 453 00:22:13,359 --> 00:22:14,949 So it's a critical point, and it's concave 454 00:22:14,950 --> 00:22:15,710 upwards around there. 455 00:22:15,710 --> 00:22:19,509 So that told us we arepoint at a minimum point, right there. 456 00:22:19,509 --> 00:22:22,430 And then at positive 3. 457 00:22:22,430 --> 00:22:24,400 So 1, 2, 3. 458 00:22:24,400 --> 00:22:26,950 At positive 3, 4.7. 459 00:22:26,950 --> 00:22:30,890 So 4.7 will look something like that. 460 00:22:30,890 --> 00:22:32,660 We have an inflection point. 461 00:22:32,660 --> 00:22:36,700 Before that, we're concave upwards, and then after that 462 00:22:36,700 --> 00:22:37,460 we're concave downwards. 463 00:22:37,460 --> 00:22:39,600 So it looks something like this. 464 00:22:39,599 --> 00:22:42,519 So we're concave upwards up with up to that point. 465 00:22:42,519 --> 00:22:45,819 Maybe, actually, you should, let me ignore that yellow 466 00:22:45,819 --> 00:22:47,549 thing I drew before. 467 00:22:47,549 --> 00:22:49,549 Let me get rid of that. 468 00:22:49,549 --> 00:22:52,930 Let me draw it like 1, 2, 3. 469 00:22:52,930 --> 00:22:59,039 3, 4.7 looks like that, and minus 3, 4.7, 1, 2, 3, 470 00:22:59,039 --> 00:23:01,289 4.7 looks like that. 471 00:23:01,289 --> 00:23:07,200 So we know at 0, we are slope of 0 and we're concave upwards, 472 00:23:07,200 --> 00:23:09,140 so we look like this. 473 00:23:09,140 --> 00:23:12,320 We're concave upwards, until x is equal to 3. 474 00:23:12,319 --> 00:23:17,689 And at x is equal to 3, we become concave downwards, and 475 00:23:17,690 --> 00:23:22,299 we go, let me try my best to draw it well, and we 476 00:23:22,299 --> 00:23:23,799 go off like that. 477 00:23:23,799 --> 00:23:28,169 And then we're concave upwards around 0, until we get, we're 478 00:23:28,170 --> 00:23:32,289 concave upwards as long as x is greater than minus 3, and 479 00:23:32,289 --> 00:23:34,440 then at minus 3 we become concave downwards again. 480 00:23:34,440 --> 00:23:35,620 Maybe I should do it in that color. 481 00:23:35,619 --> 00:23:40,009 This concave downwards right here, that's this, right here. 482 00:23:40,009 --> 00:23:41,720 That's that, right there. 483 00:23:41,720 --> 00:23:45,339 And this concave downwards, right here-- sorry, I meant 484 00:23:45,339 --> 00:23:48,439 to do it in the red color-- this concave downwards right 485 00:23:48,440 --> 00:23:51,440 here, is this, right there. 486 00:23:51,440 --> 00:23:54,490 And then the concave upwards around 0 is right there. 487 00:23:54,490 --> 00:23:57,079 You could even imagine, this concave upwards that we 488 00:23:57,079 --> 00:23:59,659 measured, that's this, concave upwards, and then this 489 00:23:59,660 --> 00:24:01,580 concave upwards is that. 490 00:24:01,579 --> 00:24:03,699 And then around 0, we're always upwards. 491 00:24:03,700 --> 00:24:07,340 So this is my sense of what the graph will look like and maybe 492 00:24:07,339 --> 00:24:10,049 it'll just you know it turns into well you could think about 493 00:24:10,049 --> 00:24:13,419 what it does is x approaches positive or negative infinity, 494 00:24:13,420 --> 00:24:15,230 some of the terms, well, I won't go into that. 495 00:24:15,230 --> 00:24:18,430 But let's test whether we've grafted correctly using 496 00:24:18,430 --> 00:24:19,570 a graphing calculator. 497 00:24:19,569 --> 00:24:25,149 So let me get out my TI-85, trusty TI-85, and let's 498 00:24:25,150 --> 00:24:26,480 graph this sucker. 499 00:24:26,480 --> 00:24:28,289 All right, press graph. 500 00:24:28,289 --> 00:24:38,829 y equals the natural log of x to the fourth plus 27. 501 00:24:38,829 --> 00:24:40,659 All right, I want to hit that graph there. 502 00:24:40,660 --> 00:24:43,650 So I do second, graph. 503 00:24:43,650 --> 00:24:45,759 And let's cross our fingers. 504 00:24:45,759 --> 00:24:47,379 It looks pretty good! 505 00:24:47,380 --> 00:24:49,770 It looks almost exactly like what we drew. 506 00:24:49,769 --> 00:24:53,680 So I think our I think our mathematics was correct. 507 00:24:53,680 --> 00:24:55,650 This was actually very satisfying. 508 00:24:55,650 --> 00:24:59,060 So hopefully you appreciate the usefulness of inflection 509 00:24:59,059 --> 00:25:02,809 points, and second derivative, and first derivative, in 510 00:25:02,809 --> 00:25:06,859 graphing some of these functions. 511 00:25:06,859 --> 00:25:07,332