Question

Is `f(x)=-2x^3-2x^2+8x-1` concave or convex at `x=3`?

Answer 1

Concave (sometimes called "concave down")

Explanation:

Concavity and convexity are determined by the sign of the second derivative of a function:

• If `f''(3)<0`, then `f(x)` is concave at `x=3`.
• If `f''(3)>0`, then `f(x)` is convex at `x=3`.

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

`f(x)=-2x^3-2x^2+8x-1`

`f'(x)=-6x^2-4x+8`

`f''(x)=-12x-4`

The value of the second derivative at `x=3` is

`f''(3)=-12(3)-4=-40`

Since this is `<0`, the function is concave at `x=3`:

These are the general shapes of concavity (and convexity):

We can check the graph of the original function at `x=3`:

graph{-2x^3-2x^2+8x-1 [-4,4, -150, 40]}