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# How do you know where the graph of f(x) is concave up and where it is concave down for f(x) = x^3 + x?

Aug 14, 2015

The graph of $f$ is concave up on intervals on which $f ' ' \left(x\right)$ is positive and the graph is concave down where $f ' ' \left(x\right)$ is negative.

#### Explanation:

So we need to investigate the sign of $f ' ' \left(x\right)$.

$f \left(x\right) = {x}^{3} + x$

$f ' \left(x\right) = 3 {x}^{2} + 1$

$f ' ' \left(x\right) = 6 x$

Clearly, $f ' ' \left(x\right) = 6 x$ is negative for $x < 0$ and positive for $x > 0$.

So the graph of $f$ is concave down on $\left(- \infty , 0\right)$ and it is concave up on $\left(0 , \infty\right)$