Question

For what values of x is `f(x)=((5x)/2)^(2/3) - x^(5/3` concave or convex?

Answer 1

`f(x)` is concave for all `x` with `x>0`

Explanation:

Writing `f(x)` in the form

`f(x)=(5/2)^(2/3)*x^(2/3)-x^(5/3)`
then

`f'(x)=(5/2)^(2/3)*2/3*x^(-1/3)-5/3*x^(2/3)`
so
`f''(x)=(5/3)^(2/3)*(-2/9)*x^(-4/3)-10/9*x^(-1/3)<0`