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

Nov 19, 2015

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 $f ' ' \left(x\right) > 0$ for all $x$ in the interval.

In this case,

$f ' ' \left(x\right) = - 40 {x}^{3} - 24 {x}^{2}$.

So, $f ' ' \left(- 2\right) = - 40 \left(- 8\right) - 24 \left(4\right) < 0$

$f ' ' \left(x\right)$ is continuous near $- 2$, so $f ' ' \left(x\right) < 0$ 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 $- 2$.)