# How do you determine the concavity for f(x) = x^4 − 32x^2 + 6?

Apr 17, 2015

The concavity of a function is the sign of its second derivative.
If, in a set, it is positive, than the concavity is up, if negative the concavity is down, if it is zero, there could be an inflection point there.

So:

$y ' = 4 {x}^{3} - 64 x$

$y ' ' = 12 {x}^{2} - 64$,

than

$12 {x}^{2} - 64 > 0 \Rightarrow {x}^{2} > \frac{64}{12} \Rightarrow {x}^{2} > \frac{16}{3} \Rightarrow$

$x < - \frac{4}{\sqrt{3}} \vee x > \frac{4}{\sqrt{3}}$, or , better:

$x < - \frac{4}{3} \sqrt{3} \vee x > \frac{4}{3} \sqrt{3}$. In this set the function has concavity up, in the complementary set it has concavity dawn, in $\pm \frac{4}{3} \sqrt{3}$ there are two inflection points.