Question

For what values of x is `f(x)=3x^3-7x^2-5x+9` concave or convex?

Answer 1

`f` is concave (concave down) on `(-oo,7/9)` and is convex (concave up) on `(7/9,oo)`.

Explanation:

The convexity and concavity of the function `f` can be determined by looking at the sign of the second derivative:

• If `f''>0`, then `f` is convex.
• If `f''<0`, then `f` is concave.

To find the function's second derivative, use the power rule.

`f(x)=3x^3-7x^2-5x+9`

`f'(x)=9x^2-14x-5`

`f''(x)=18x-14`

So, the convexity and concavity are determined by the sign of `f''(x)=18x-14`.

The second derivative equals `0` when `18x-14=0`, which is at `x=7/9`.

When `x>7/9`, `f''(x)>0`, so `f(x)` is convex on `(7/9,oo)`.

When `x<7/9`, `f''(x)<0`, so `f(x)` is concave on `(-oo,7/9)`.