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Georgia Southern University

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Problem 48

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.

$$

k(x)=x^{3}+3 x^{2}+3 x+1, \quad-\infty<x \leq 0

$$

Answer

$x=0$

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## Discussion

## Video Transcript

all right here we have care. Bax is equal to x cubed plus three x squared plus three x close one. Well, this is equal to X plus three x plus one Cute Nazis. Think you know that's just a simple binomial expansion. So the derivative that makes the derivative a little bit nicer comes pre factored. It's three times X plus one squared. So where does K Prime nickel zero o? Let's go ahead and into the domain restriction years. So let's go in for that. And so we have negative Infinity is less an ex Western Eagle two zero. And so where this K prime equals zero, well, that's happens when Nexus negative one. Okay, so what's look a k prime? But this is pretty easy because we can notice that K prime is always greater than or equal to zero. So the plot are a critical point. We're stopping in zero k. Prime is going to be positive, and I'm positive. So it's going up on leveling off increasing. So you actually see what's gonna happen Way have one local max to the local necks at zero, with the value of explaining zero, we get one and this is the absolute max occurs at X equals zero because this function is just increasing all the way up to zero and stop sincere. And there's committee no absolute men, because case coming for minus infinity.

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