# For what values of x is f(x)= -2x^3 + 4 x^2 + 5x -5  concave or convex?

Dec 10, 2017

Concave: $\left\{x \in \mathbb{R} : \frac{2}{3} < x < \infty\right\}$

Convex: $\left\{x \in \mathbb{R} : - \infty < x < \frac{2}{3}\right\}$

#### Explanation:

A function is convex if the second derivative is positive and concave where its second derivative is negative. When the second derivative is $0$ this could mean the function is concave , convex, or it could be a point of inflexion. This would have to be tested using the first derivative.

$f \left(x\right) = - 2 {x}^{3} + 4 {x}^{2} + 5 x - 5$

$\frac{\mathrm{dy}}{\mathrm{dx}} \left(- 2 {x}^{3} + 4 {x}^{2} + 5 x - 5\right) = - 6 {x}^{2} + 8 x + 5$

$\frac{{d}^{2} y}{{\mathrm{dx}}^{2}} \left(- 6 {x}^{2} + 8 x + 5\right) = - 12 x + 8$

So:

$- 12 x + 8 < 0$

$x > \frac{8}{12}$

$x > \frac{2}{3}$

Concave: $\left\{x \in \mathbb{R} : \frac{2}{3} < x < \infty\right\}$

$- 12 x + 8 > 0$

$x < \frac{8}{12}$

$x < \frac{2}{3}$

Convex: $\left\{x \in \mathbb{R} : - \infty < x < \frac{2}{3}\right\}$

Graph:

graph{y=-2x^3+4x^2+5x-5 [-4, 4, -7.12, 7.12]}