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

Jan 29, 2017

The answer is $f \left(x\right)$ is concave down for x in ]-oo, 1] and concave up when  x in [1, +oo[

#### Explanation:

We calculate the first derivative and we build a sign chart

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

$f ' \left(x\right) = 20 {x}^{4} - 20 {x}^{3}$

$f ' \left(x\right) = 20 {x}^{3} \left(x - 1\right)$

The critical points are when $f ' \left(x\right) = 0$

$20 {x}^{3} \left(x - 1\right) = 0$

$x = 0$ and $x = 1$

Now we construct the sign chart

$\textcolor{w h i t e}{a a a a}$$x$$\textcolor{w h i t e}{a a a a}$$- \infty$$\textcolor{w h i t e}{a a a a a a a}$$0$$\textcolor{w h i t e}{a a a a a a}$$1$$\textcolor{w h i t e}{a a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$x$$\textcolor{w h i t e}{a a a a a a a a}$$-$$\textcolor{w h i t e}{a a}$color(white)(aaaa)+$\textcolor{w h i t e}{a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$x - 1$$\textcolor{w h i t e}{a a a a a}$$-$$\textcolor{w h i t e}{a a a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$+$

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

$\textcolor{w h i t e}{a a a a}$$f \left(x\right)$$\textcolor{w h i t e}{a a a a a a}$↗^(0)$\textcolor{w h i t e}{a a a a}$↘_(-1)$\textcolor{w h i t e}{a a a a}$↗^(+oo)

Therefore,

$f \left(x\right)$ is concave down for x in ]-oo, 1] and concave up when

 x in [1, +oo[