8 multiple choice 1 point for which values of a and b does...

Name: 8 multiple choice 1 point for which values of a and b does the function fx fracaxbx2 have a critical point at x 2 and x 2 a 1 b 4 a 1 b 1 a 2 b 2 a 4 b 2 a 1 b 3 clear my selection 29927
Uploaded: 2021-05-21T21:34:35-08:00
Duration: 7 min 56 s
Channel: Nick Johnson
Description: 8 multiple choice 1 point for which values of a and b does the function fx fracaxbx2 have a critical point at x 2 and x 2 a 1 b 4 a 1 b 1 a 2 b 2 a 4 b 2 a 1 b 3 clear my selection 29927

8 Multiple Choice 1 point For which values of a and b does the function $f(x) = \frac{ax}{b+x^2}$ have a critical point at $x = 2$ and $x = -2$? $a = 1, b = 4$ $a = 1, b = 1$ $a = 2, b = 2$ $a = 4, b = 2$ $a = 1, b = 3$ Clear my selection

Transcript

0:00 Hello.

00:01 So here we have the function f of x is equal to 2x squared over x to the fourth plus 1.

00:06 And in part a, we want to determine is the point negative 1 -1 on the graph of f -fx? well, if it is, that would mean that f of negative 1 would be equal to 1.

00:19 When the input is negative 1, the output is 1.

00:22 Well, is it? well, so what is, right, what is f of negative 1? well, f of negative 1, that would be equal to, well, 2 times negative 1 squared over negative 1 to the 4th plus 1.

00:42 Well, this is equal to 2 times 1 or 2 over 1 plus 1 or 2.

00:48 So 2 over 2 is equal to 1.

00:51 So what do we have? we have that f of negative 1 is equal to 1.

00:58 So then yes, right, the point negative 1, 1 is going to be on the graph of f of x.

01:05 Then for part b, well, if x is equal to 2, then we have f of 2.

01:14 So f of 2 is going to be equal to 2 times 2 squared over 2 to 2 to 2.

01:24 The fourth plus one.

01:27 So that's equal to two times four, which is eight over, while two to the fourth is two times two times two times two.

01:35 That's 16 plus one, which is 17.

01:38 So we have that f of two is equal to eight 17th.

01:42 So therefore, we have the point two comma eight 17th is on the graph, is on the graph of f of x.

01:52 Okay, and then for part c, what do we have? well, if f of x is equal to 1, then the function is equal to 1.

02:02 So that means the function is 2x squared over x to the 4th plus 1.

02:09 We set this equal to 1, and then it's out for x.

02:13 So we multiply both sides by x to the 4th plus 1, to clear the fractions, and then we end up with just 2x4.

02:20 Squared, which is 2x squared, is then equal to, well, x to the fourth plus one, we can then set this equal to zero to get x to the fourth minus 2x squared plus one is then equal to zero.

02:41 We can then factor this as, well, we have zero is equal to, this is in a factor as x, this is difference of two squares, basically.

02:51 This is x squared minus 1 times x squared plus 1.

02:56 We can always check by distributing and get back to x to the 4th minus 2x squared plus 1.

03:01 But then we can factor this further because, i mean, x squared plus 1 cannot factor anymore, but the difference of two squares can, yes.

03:10 So x squared minus 1 can factor further.

03:13 So this is going to be 0 is equal to x squared minus 1.

03:16 That factors as x plus 1 times x minus 1.

03:19 So we have x plus 1 times x minus 1 and then times x squared plus 1.

03:26 Okay, so now we have three factors equal to 0.

03:29 So that means, you know, any of these factors can be equal to 0.

03:33 Either x plus 1 is equal to 0 or x minus 1 equal to 0 or x squared plus 1 equal to 0.

03:40 Well, if x plus 1 is equal to 0, that implies that x is equal to negative 1, or if x minus 1 is equal to 0, that implies that x is equal to positive 1.

03:52 Or, well, if x squared plus 1 is equal to 0, well, wait a minute.

03:56 That's never true because, i mean, you could try to solve this, but x squared is always positive.

04:02 Regardless, if x is positive or negative, right, it could be 0, but then 0 plus 1 is 1, right? but if x is negative or positive, x squared is always positive, and we're adding 1.

04:12 So that is never going to be equal to zero.

04:16 You only have two values that make this true, either negative one or one...

Question

8 Multiple Choice 1 point For which values of a and b does the function $f(x) = \frac{ax}{b+x^2}$ have a critical point at $x = 2$ and $x = -2$? $a = 1, b = 4$ $a = 1, b = 1$ $a = 2, b = 2$ $a = 4, b = 2$ $a = 1, b = 3$ Clear my selection

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