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

Mar 8, 2017

$f \left(x\right)$ is concave when x in ]-oo,25/18[ and convex when x in ]25/18,+oo[

#### Explanation:

We develop the expression and we calculate the first and second derivatives

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

$= \left(6 {x}^{2} - 13 x + 5\right) \left(x - 2\right)$

$= 6 {x}^{3} - 12 {x}^{2} - 13 {x}^{2} + 26 x + 5 x - 10$

$f \left(x\right) = 6 {x}^{3} - 25 {x}^{2} + 31 x - 10$

$f ' \left(x\right) = 18 {x}^{2} - 50 x + 31$

$f ' ' \left(x\right) = 36 x - 50$

$f ' ' \left(x\right) = 0$ when $x = \frac{50}{36} = \frac{25}{18}$

We construct a table

$\textcolor{w h i t e}{a a a a}$$I n t e r v a l$$\textcolor{w h i t e}{a a a a}$$|$$\textcolor{w h i t e}{a a a a}$]-oo,25/18[$\textcolor{w h i t e}{a a a a}$$|$$\textcolor{w h i t e}{a a a}$]25/18,+oo[$|$

$\textcolor{w h i t e}{a a a a}$$S i g n f ' ' \left(x\right)$$\textcolor{w h i t e}{a a}$$|$$\textcolor{w h i t e}{a a a a a a a a}$$-$$\textcolor{w h i t e}{a a a a a a a}$$|$$\textcolor{w h i t e}{a a a a a a a a}$$+$$\textcolor{w h i t e}{a a}$$|$

$\textcolor{w h i t e}{a a a a}$$f u n c t i o n$$\textcolor{w h i t e}{a a a a}$$|$$\textcolor{w h i t e}{a a a a a a a a a}$$\bigcap$$\textcolor{w h i t e}{a a a a a a}$$|$$\textcolor{w h i t e}{a a a a a a a a}$$\bigcup$$\textcolor{w h i t e}{a a}$$|$

Therefore,

$f \left(x\right)$ is concave when x in ]-oo,25/18[ and convex when x in ]25/18,+oo[