# On what interval is f(x)=6x^3+54x-9 concave up and down?

Apr 20, 2015

A function is concave up when the second derivative is positive, it is concave down when it is negative, and there could be an inflection point when it is zero.

$y ' = 18 {x}^{2} + 54$

$y ' ' = 36 x + 54$

so:

$y ' ' > 0 \Rightarrow x > - \frac{54}{36} \Rightarrow x > - \frac{3}{2}$.

In $\left(- \frac{3}{2} , + \infty\right)$ the concave is up,

in $\left(- \infty , - \frac{3}{2}\right)$the concave is down,

in $x = - \frac{3}{2}$ there is an inflection point.