Question

Is `f(x)=-2x^5+7x^2-6x+3` concave or convex at `x=-4`?

Answer 1

Convex.

Explanation:

To test a function's concavity or convexity, find the value of the second derivative of the function at a point:

• If `f''(-4)<0`, then `f(x)` is concave at `x=-4`.
• If `f''(-4)>0`, then `f(x)` is convex at `x=-4`.

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

`f(x)=-2x^5+7x^2-6x+3`

`f'(x)=-10x^4+14x-6`

`f''(x)=-40x^3+14`

The value of the second derivative at `x=-4` is

`f''(-4)=-40(-4)^3+14=-40(-64)+14=2574`

Since this is `>0`, the function is convex at `x=-4`.