Example 1 | Wiskunde op Tilburg University

Consider the function

$y(x)=x^3+2x$ . It holds that

$y'(x)=3x^2+2$ ;
$y''(x)=6x$ .

It follows that

$y''(x)\leq 0$ for every $x\leq 0$ ;
$y''(x)=0$ for $x=0$ ;
$y''(x)\geq 0$ for every $x\geq 0$ .

Conclusion:

$y(x)$ is concave on the interval

$(-\infty,0]$ ,

$y(x)$ is convex op the interval

$[0,\infty)$ and

$x=0$ is an inflection point.