Critical Points

Question

Determine all the critical points for the cubed root of (t squared) times (2t-1):

(cubed root of (√t^2)) (2t−1)

A step by step explanation would be much appreciated

Answer 1

In my opinion the easiest way to do this is to expand the function as

t^2/3(2t-1) = 2t^5/3 - t^2/3

A graph will show that there are only 2 critical points and they should be able to be found by setting the derivative = 0.

Nope, I am wrong.

Compute the derivative by the product rule.

f'(t) = 2t^2/3 + (2t-1)(2/3)t^-1/3 = 0

This has 2 solutions: t=0 and (if you multiply through by t^1/3, you will easily see that the other critical point is at t =1/5.