Question

For what values of x is `f(x)=4x^3-3x+5` concave or convex?

Answer 1

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

Explanation:

Convexity and concavity are determined by the sign of the second derivative.

• If `f''(a)>0`, then `f(x)` is convex at `x=a`.
• If `f''(a)<0`, then `f(x)` is concave at `x=a`.

Find the second derivative of the function.

`f(x)=4x^3-3x+5`
`f'(x)=12x^2-3`
`f''(x)=24x`

Analyze the sign of the second derivative, `24x`.

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

Thus,

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