Question

For what values of x is `f(x)= 24x^3-12x` concave or convex?

Answer 1

Convex on `(0,oo)`; concave on `(-oo,0)`

Explanation:

The concavity and convexity of a function are determined by the sign (positive or negative) of a function's second derivative.

Hence, to determine when a specific function is concave or convex, we first must find its second derivative.

Through the power rule, we see that

`f(x)=24x^3-12x`

`f'(x)=72x^2-12`

`f''(x)=144x`

We can clearly see in `144x` that this is positive when `x>0` and negative when `x<0`.

In order to apply concavity/convexity, use the definitions:

• `f(x)` is convex when `f''(x)>0`.
• `f(x)` is concave when `f''(x)<0`.

Thus,

• `f(x)` is convex on `(0,oo)`.
• `f(x)` is concave on `(-oo,0)`.