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Georgia Southern University

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Problem 46

a. Identify the function's local extreme values in the given domain, and say where they occur.

b. Which of the extreme values, if any, are absolute?

c. Support your findings with a graphing calculator or computer grapher.

$$

f(t)=t^{3}-3 t^{2}, \quad-\infty< t \leq 3

$$

Answer

$(a)[\text { local maximum at } t=0)(b)[\text { absolute maximum at } t=0,3$

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## Discussion

## Video Transcript

Okay, So you have. Ah, a tea. It's tick you to minus three. T squared. Santee, between your infinity hand three says, take the derivative to find our critical points. We get three t squared minus sixteen instead of that crime zero. We have that, uh, in fact, two out of three T three tee times T minus two zero. So either t zero tea is too stroller number line. You have zero and two, and then the end of our domain, which is three. So it's like a deaf prime Texas less than zero. This is negative and negatives of positive if, uh, f promise between your into this is negative, but this is positive and negative. And if x is greater than to have positive. So our function is going like that and stopping at three. So we see that we have a local max where well, at zero zero and three zero be plugging three, begin zero zero zero, and then we have a local men, and okay, Tio, what do we get? Money plucking too. Eight minus twelve. So negative for there's not going to be an absolute men because the functions coming from minus infinity. But the absolute max is going to be zero and occurs in X equals zero and X equals three