Question

there are no examples of this type of problem in my book so if you could help walk me through it - that would be extremely helpful. thanks ahead of time.

Find the extreme values of the function on the interval and where they occur.
4) F(x)=³√(x); -3</=x</=64
A. Maximum at (64, 4), and minimum at (-3, ³√-3)
B. Maximum at (-64, 4), and minimum at (0,0)
C. Maximum at (0,0), and minimum at (64,4)
D. Maximum at (64,4), and minimum at (-64,-4).

Thanks again.

Answer 1

There must be problems on extrema. The extrema occur where f' = 0. Here,

f(x) = x^(1/3)
f'(x) = 1/3 x^(-2/3)
f'(x) is never zero anywhere, especially in this interval.

So, we just want the maximum and minimum values attained on the interval. Note that f' > 0 everywhere, so it is strictly increasing. That means

f(-3) < f(64)
f(-3) = -∛3
f(64) = 4
So, those are the extrema on this interval.

Looks like (A) to me.

Answer 2

Thanks! I'll go with that and then if I do get it wrong - I'll ask for an explanation as to why from my teacher. Thanks for explaining it too :) It's starting to make more sense.