Question

Is `f(x)=-2x^5-3x^4+15x-4` concave or convex at `x=-4`?

Answer 1

Convex (sometimes called "concave upwards").

Explanation:

The concavity and convexity of a function can be determined by examining the sign of a function's second derivative.

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

Note that: you may call concave "concave down" and convex "concave up."

We must find the function's second derivative through the power rule:

`f(x)=-2x^5-3x^4+15x-4`

`f'(x)=-10x^4-12x^3+15`

`f''(x)=-40x^3-36x^2`

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

`f''(-4)=-40(-4)^3-36(-4)^2=1984`

Since this is `>0`, the function is convex (sometimes called concave up) at `x=-4`.