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Numerade Educator



Problem 40 Medium Difficulty

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.$$
k(x)=x^{2 / 3}\left(x^{2}-4\right)


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Video Transcript

Okay, here we have chaos. Max Neagle's X to the two thirds times X squared minus four. Let's take the derivative to find our critical points. So we have two thirds x to the minus one third times X squared minus four Worse X to the two thirds times to X. And again I'm gonna multiply the top and bottom by three x to the one third to get a common denominator. Okay, so that's two Next to the one turn Third Time's extra minus one third is no sorry. I said this already has three extra one third and denominator, this's just two times X squared minus four. Plus I'm gonna multiplied by three is getting a six. You know, I want to buy X to the one third times x to the two thirds of x times x x squared can all over three next to the one third Great. And so Windows K prime equals zero. Well, that's when looks like and X squared minus eight. Syria or exit is plus reminds one. And in Cape Prime is undefined. That's the numerator zero with the denominator, zero k prime will be undefined the next zero. So we have critical points. That one negative one and zero. So thinking of one Ciro one. If we look a caped crime to the left of negative one and to the right of one, the numerator will be positive. And so here the nominators Negative. So have negative, President. And then the exact opposite Here the denominator will be negative. Your anger will be positive. The nominee will be positive. See that? Right. So here the denominator, we'LL be negative and the new railroad B negative. And in here and numerator will be negative in the denominator will be positive. Open. So we see that Kay is decreasing evening between negative infinity and negative one increasing between negative one and zero decreasing between zero and one and then increasing between one divinity and so we have some extreme. We have a local man at negative one of what? Actually, we have local men's at plus or minus one Thanks And they both end up having the same value of ninety three. Okay, in any of the local max, they're zero. We have no absolute max because the functions going off to infinite positive Infinity to the left, to the right. But we do have a local men of negative three and that occurs at plus or minus one