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Georgia Southern University

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Problem 22

a. Find the open intervals on which the function is increasing and decreasing.

b. Identify the function's local and absolute extreme values, if any, saying where they occur.

$$

h(x)=2 x^{3}-18 x

$$

Answer

$$

-\sqrt{3}, 1

$$

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## Discussion

## Video Transcript

All right, look. So we have h of X is to execute minus eighteen. X was take the derivative because we're going to want to find our critical points. Six X squared, minus eighteen of H crime is zero. We get that well, six x squared. His eighteen or X is plus or minus three. There's divided by six and six square you up a plus and minus square root. So the intervals for which H is increasing or decreasing Well, pull up a critical points minus route three. And plus you are three. Okay, If we look at crime, if X is less than minus room three well, then this factor here is going to be greater than eighteen. So we're gonna be positive if X is between negative Route three hundred three, say zero. The promise. Khun B. Negative. I guess. Technically, this is a tch prime and then effects that's created the Route three h prime is going to be positive. So you see, the age is increasing from minus infinity two minus three three. It's decreasing from minus three three to room three, and it's increasing from room three to infinity. Of course, it's increasing exactly where the derivative is positive is decreasing. Exactly real driven. It's negative. Okay? And then we see our extreme. We have the local max or a changes from being increasing, decreasing to that snagged Route three. So our local max is going to be minus route three and it has a value of two or three, and then we have a little cold man on Route three with a value of minus territory. You just look at him and they're no absolute max or groups of men because the function is going to be increasing and increasing.