Find the absolute maximum and minimum values of the...

Find the absolute maximum and minimum values of the following function on the given interval. Then graph the function. g(x) = sqrt(9 - x^2), -3 <= x <= 2 Find the absolute maximum value. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The absolute maximum value occurs at x = . (Type exact answers, using radicals as needed. Use a comma to separate answers as needed.) B. There is no absolute maximum.

Transcript

00:01 In this problem, we have this function, g of x.

00:04 Oops, let me make sure that looks like a g.

00:05 G of x equals the square root of 9 minus x squared.

00:11 Now, we need to find the maximum and minimum values of this function.

00:16 And then grab it.

00:16 Now, the way that we're going to do that is we're going to take the first derivative and then take the second derivative.

00:22 Now, we're going to take the first derivative and make that equal to zero and then find our critical points.

00:28 So the first thing i'm going to do is rewrite this.

00:30 As an exponent, 9 minus x squared to the 1 half.

00:34 Let me show you, looks up a 2, right, to the 1 half, to the 1 half.

00:38 That we don't make that look very nice, can i? there we go.

00:41 Now, i'm going to use the power rule, g of x, and i'm actually going to use the power roll in the chain rule.

00:47 So i'm going to say that this is equal to 1 ,5, 9 minus x squared to the negative 1⁄2, and then i'm going to multiply that times the derivative of the inner function, which is going to be negative 2x.

01:02 All right.

01:02 Now, simplifying one half of negative 2 is negative 1.

01:05 So this is going to be negative x.

01:09 I'll say negative x.

01:12 Well, this is how i'll do it first, negative x over the square root of 9 minus x squared.

01:19 Now, i'm doing that because i want to set this equal to zero.

01:22 And that's actually going to be quite easy.

01:24 That means that x is going to equal zero because i have to multiply by 9 minus x squared and i'm also going to find that x is equal to zero.

01:36 So where x equals zero is a critical point.

01:40 Now we also have to think of the domain of this function, right? the domain is anywhere between positive and negative three because we can't take the square root of a negative number and it be in the real numbers.

01:53 So here we also have that x in the domain of this, x is not differentiable where x equals three or negative three because that would mean that we have a denominator of zero.

02:07 So x is not differentiable here at plus or minus three.

02:12 All right.

02:12 No, that's important because those are test points that we need for our minimum and maximum.

02:18 Now, we're going to take the second derivative now.

02:21 So i'm going to keep this as, now i'm going to rewrite this, i just say, in order to make it easier to differentiate.

02:27 So g prime of x is going to equal negative x times 9 minus x squared to the negative 1 half.

02:38 All right.

02:38 Now i'm just going to draw a little dotted line here to separate my answers.

02:42 All right, so that now we're looking here and keep it a little more orderly for us.

02:45 Now, we're going to do the same process.

02:48 We're going to use the power rule and the chain rule...

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