# For what values of x is f(x)=(5x-1)(x-5) (2x+3) concave or convex?

Jan 13, 2018

Convex $\textcolor{w h i t e}{888} x \in \left(\frac{37}{30} , \infty\right)$

Concave $\textcolor{w h i t e}{888} x \in \left(- \infty , \frac{37}{30}\right)$

#### Explanation:

We can test for concavity using the second derivative. If:

$\frac{{d}^{2} y}{{\mathrm{dx}}^{2}} > 0$ convex ( concave up )

$\frac{{d}^{2} y}{{\mathrm{dx}}^{2}} < 0$ concave ( concave down )

$\frac{{d}^{2} y}{{\mathrm{dx}}^{2}} = 0$ concave/convex or point of inflection. This would have to be tested.

$f \left(x\right) = \left(5 x - 1\right) \left(x - 5\right) \left(2 x + 3\right)$

It will make the differentiation easier if we expand this:

$\left(5 x - 1\right) \left(x - 5\right) \left(2 x + 3\right) = 10 {x}^{3} - 37 {x}^{2} - 68 x + 15$

The second derivative is the derivative of the first derivative, so:

$\frac{\mathrm{dy}}{\mathrm{dx}} \left(10 {x}^{3} - 37 {x}^{2} - 68 x + 15\right) = 30 {x}^{2} - 74 x - 68$

$\frac{{d}^{2} y}{{\mathrm{dx}}^{2}} = \frac{\mathrm{dy}}{\mathrm{dx}} \left(30 {x}^{2} - 74 x - 68\right) = 60 x - 74$

$\therefore$

$60 x - 74 > 0$ , $x > \frac{37}{30}$

Convex $\textcolor{w h i t e}{888} x \in \left(\frac{37}{30} , \infty\right)$

$60 x - 74 < 0$ , $x < \frac{37}{30}$

Concave $\textcolor{w h i t e}{888} x \in \left(- \infty , \frac{37}{30}\right)$

GRAPH: