let f be the function defined by f(x)=12x^2/3 -4X

Question

a)find the intervals on which f is increasing
I got the rest of the question but I'm still not getting the increasing/decreasing thing

Answer 1

As we saw in a previous posting, f(x) is increasing where f'(x) is positive.

A positive slope means that when x increases, f(x) increases. A negative slope means that as x increases, f(x) decreases.

f'(x) = 8x^(-1/3) - 4

So, we want 8/x^1/3 - 4 > 0
8/x^1/3 > 4
1/x^1/3 > 1/2
x^1/3 > 2
x > 8

Answer 2

Oops. Taking reciprocals reverses the direction of the inequality:

8/x^1/3 - 4 > 0
8/x^1/3 > 4
1/x^1/3 > 1/2
x^1/3 < 2
x < 8