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Problem 12

Answer the following questions about the functions whose derivatives

are given.

a. What are the critical points of $f ?$

b. On what open intervals is $f$ increasing or decreasing?

c. At what points, if any, does $f$ assume local maximum and minimum values?

$$

f^{\prime}(x)=x^{-1 / 2}(x-3)

$$

Answer

see the graph

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## Discussion

## Video Transcript

all right. So derivative or function is given by OK, X to the minus one half of her ex mines three bomb is going to write that his ex minus three over square root of X. Good. Okay. And so if we're looking for a critical points well, we see the X equals three is going to make f prime undefined. And then also, X equals zero X equals three is going to make a private call zero Sorry. And then X equals zero is gonna make of prime on to find. So we need to consider those values. Okay. And then we want to know where is f increasing and decreasing. Well, fans era and three, we can look enough crime to see where l promise positive or negative Tell us where is increasing or decreasing. So less than zero f crime was going to be undefined because we'll be taking the square root of a night, remember? And then between zero and three it looks like the derivative is negative. Good morning. And then, um let's see between, uh, for ex greater than three, the derivative is going to be positive. And so it looks like our function no. Okay, he's going to start at zero and it's going to be decreasing between zero and three and then an increasing from three to infinity and then or extreme A. It looks like we have a local man and X equals three and then no local Max because it never. The function never changes from being increasing, decreasing and only changes from seem decreasing to increasing it through It's a local man.

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