(a) Find the critical points, if any, of the following...

(a) Find the critical points, if any, of the following function on the given interval. (b) Determine the absolute extreme values of f on the given interval. (c) Use a graphing utility to confirm your conclusions. $\frac{2}{3}$ f(x) = (x + 2)$^{\frac{2}{3}}$ on [-10,6] (a) Find the critical points, if any, of f(x) on [-10,6). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The critical point(s) occur(s) at x = (Use a comma to separate answers as needed.) B. There are no critical points for f(x) on [-10,6].

Transcript

00:01 In this problem, we are given the function f of theta equals 6 sine theta plus 2 cosine theta on the interval from negative 2 pi to 2 pi.

00:08 Our job is to find the critical points on that interval, to identify the local maximum and minimum points, and to determine the absolute maximum and minimum values.

00:17 So we start on critical points by finding the critical numbers, which means we have to take the derivative.

00:22 So we've taken the derivative of the given function.

00:25 That derivative is 6 cosine theta minus 2 sine theta.

00:29 That exists for all real numbers theta, so the critical numbers occur when the derivative is zero.

00:35 So we're going to solve this equation for values of theta in the original interval.

00:43 And so we see that we want all values of theta in that interval, where tangent of theta equals three.

00:50 One of those values is tangent inverse of three, which turns out to be approximately 1 .249.

01:03 To get the others, we will add and subtract values of pi.

01:09 And so we're going to have four values in all.

01:13 We will give the approximations for those here.

01:18 So adding and subtracting pi and staying within that interval, we see that each of these four values rounded to three decimal places will be a critical number in the interval.

01:36 Now, once we get those, we want to find the y coordinates that correspond to these theta coordinates.

01:43 And so to do that, we have f of theta values in each case.

01:55 We'll just write these underneath.

01:57 We're going to sketch the graph in just a second...

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