# Thread: Where is f(x) concave and convex?

1. ## Where is f(x) concave and convex?

Hello!

I have this function $f(x)=1,2x^5-4x^3$ and I need to give the intervals where the function's graph is concave and convex.

I know $f'(x)=0$ when $x=\sqrt{2}$, $x=-\sqrt{2}$ and $x=0$

2. From what I understand of concave and convex is as follows:

Concave: Points on a graph where the slope is decreasing.
Convex: Points on a graph where the slope is increasing.

This of course is a just a simple explanation, like I said, from what I understand.

So I'd say your looking for all the points where:
$f''(x) > 0$ These would be the convex set.
$f''(x) < 0$ These would be the concave set.

3. $f''(x) = 24x^3 - 24x = 24x(x^2-1) \Rightarrow f''(x) = 0$ at $x=0, x=1, x=-1$

$f''(2) = 24(8-2)$ and $f''(-2) = 24(2-8)$ , and $f(x)$ is continuous in $\mathbb{R}$ so:

$f(x)$ is concave at: $x > 1$ , $-1 < x < 0$ and convex at: $x < -1$ , $0

Woops, late :X