Question

Is `f(x)=-2x^5-2x^4+5x-45` concave or convex at `x=-2`?

Answer 1

A function (or its graph) can be said to be concave or convex on an interval. This function is convex near `-2`. (In some open interval containing `-2`.)

Explanation:

A necessary and sufficient condition for `f` to be convex on an interval is that `f''(x) >0` for all `x` in the interval.

In this case,

`f''(x) = -40x^3-24x^2`.

So, `f''(-2) = -40(-8)-24(4) <0`

`f''(x)` is continuous near `-2`, so `f''(x) <0` for `x` near `-2` and `f` is convex near `-2`.

(If you have been given a definition of `f` is "convex at a number, `a`", then I would guess you'll say `f` is convex at `-2`.)