Question

Find the absolute extrema of...

Answer 1

\[\sqrt[3]{x}\]

Answer 2

I already solved for the derivative and got... \[\frac{ 1 }{ 3 }x^{-\frac{ 2 }{ 3 }}\]

Answer 3

I'm just not exactly sure what to do next.

Answer 4

by the Weierstrass theorem we need of a closed interval of the real line

Answer 5

I can't say that I've learned about the Weierstrass theorem.

Answer 6

Is that where you just find all the critical points of the closed interval and determine which is the max?

Answer 7

yes!

Answer 8

Awesome!
One question though, how would I find the critical points of my derivative since I can't really isolate x (at least from what I can see)?

Answer 9

your first derivative is always positive, which means that your function is an increasing function

Answer 10

of course at the point \(x=0\) your first derivative is not defined

Answer 11

So it would have an asymptote?

Answer 12

no, since the asymptotes occurs at point \(x\) such that the function (not the first derivative) becomes an infinity

Answer 13

Okay, I think I understand.

Answer 14

ok! :)

Answer 15

Since my function is an increasing function, is there a way to have specific critical points? (like 1,2,3..)

Answer 16

Or would it just be infinity?

Answer 17

as I said before, we need of a closed interval. For example if we have this interval: \([2,3]\)
then we have to evaluate \(f(2),f(3)\). We will conclude that \(f(2)\) is the point of minimum, and \(f(3)\) is the point of maximum for function \(f\)

Answer 18

Okay. Got it. Thank you! :)

Answer 19

:)