# How do you find the maximum, minimum and inflection points and concavity for the function y=x^4-8x^3?

May 14, 2018

#### Explanation:

First calculate, the first derivative

$y = {x}^{4} - 8 {x}^{3}$

$y ' = 4 {x}^{3} - 24 {x}^{2} = 4 {x}^{2} \left(x - 6\right)$

The critical points are when $y ' = 0$

$4 {x}^{2} \left(x - 6\right) = 0$

$\implies$, $\left\{\begin{matrix}x = 0 \\ x = 6\end{matrix}\right.$

Build a variation 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}$$0$$\textcolor{w h i t e}{a a a a a a}$$6$$\textcolor{w h i t e}{a a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$${x}^{2}$$\textcolor{w h i t e}{a a 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 a}$$+$

$\textcolor{w h i t e}{a a a a}$$x - 6$$\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 a}$$+$

$\textcolor{w h i t e}{a a a a}$$y '$$\textcolor{w h i t e}{a a a 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 a}$$+$

$\textcolor{w h i t e}{a a a a}$$y$$\textcolor{w h i t e}{a a a 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 a}$↗

There is a local minimum at $\left(6 , - 432\right)$

Calculate the second derivative

$y ' ' = 12 {x}^{2} - 48 x = 12 x \left(x - 4\right)$

The inflection points are when $y ' ' = 0$

$12 x \left(x - 4\right) = 0$

$\implies$, $\left\{\begin{matrix}x = 0 \\ x = 4\end{matrix}\right.$

Build a variation chart

$\textcolor{w h i t e}{a a a a}$$\text{ Interval }$$\textcolor{w h i t e}{a a a a}$$\left(- \infty , 0\right)$$\textcolor{w h i t e}{a a a a}$$\left(0 , 4\right)$$\textcolor{w h i t e}{a a a a a a}$$\left(4 , + \infty\right)$

$\textcolor{w h i t e}{a a a a a a}$$\text{Sign y'' }$$\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}$$-$$\textcolor{w h i t e}{a a a a a a a a a}$$+$

$\textcolor{w h i t e}{a a a a a a}$$\text{ y }$$\textcolor{w h i t e}{a a a a a a a a a a a}$$\cup$$\textcolor{w h i t e}{a a a a a a a}$$\cap$$\textcolor{w h i t e}{a a a a a a a a a}$$\cup$

The inflection points are $\left(0 , 0\right)$ and $\left(4 , - 256\right)$

The curve is convex when $x \in \left(- \infty , 0\right) \cup \left(4 , + \infty\right)$

The curve is concave when $x \in \left(0 , 4\right)$

graph{x^4-8x^3 [-25.66, 25.67, -12.83, 12.83]}