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Georgia Southern University

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Problem 38

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.

$$

g(x)=x^{2 / 3}(x+5)

$$

Answer

See the graph

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## Discussion

## Video Transcript

All right. See you. Becks equals X to the two thirds times X plus five. Okay, so the derivative, it's gonna be two thirds next to the minus two third. Who next? Tonight, it's one third times X plus five plus X to the two thirds. And I'm gonna multiply the top bottom by X to the one third, actually, three times x to the wondered. So get two times X plus five plus just, uh, three x all over three X to the one third. So to find a critical points, where does cheap Prime equals? Zero. We want to solve two x plus ten plus three x zero R five x this planet zero or nexus Negative too. And then where's Ji Prime? Undefined. Well, that's easy. That's just all next zero. Okay, so find the intervals there. Jeez, Increasing or decreasing so well put. Negative. Two zero. I've been virtually crime When city. So here's our numerator When x is less the negative to the numerator is negative And the denominators negatives of executive and then we go between native two and zero the the numerator becomes positive, but the denominator is still negative. Zits. Come on listen. Sorry when X is less than a year to their both. Negative. So that's a positive betweennegative two and zero. The numerator is positive in the denominator is negative and then the next is greater than zero. They're both positive. So, g, it's going to be increasing from minus infinity to minus two, decreasing from negative to zero and then increasing from zero to infinity. Good. And so, Lou onesie, We have a local max Where at see negative two. Right, Because the function is changing from increasing decreasing, and the value is going to be three Cambridge for all right. Three pussy, three cube root of yeah, you two for good. And then we have a local men at zero here, and there's gonna be no absolute max syrups that men, because the function is increasing from mega infinity and then increasing to infinity

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