# How do you find all critical point and determine the min, max and inflection given f(x)=x^4-4x^3+20?

Jul 6, 2018

#### Explanation:

Calculate the first and second derivatives

The function is

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

Calculate the first derivative

$f ' \left(x\right) = 4 {x}^{3} - 12 {x}^{2}$

$f ' \left(x\right) = 0$

$\implies$, $4 {x}^{3} - 12 {x}^{2} = 0$

$\implies$, $4 {x}^{2} \left(x - 3\right) = 0$

The critical points are $x = 0$ and $x = 3$

Construct 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}$$3$$\textcolor{w h i t e}{a a a a}$$+ \infty$

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

There is a local minimum at $\left(3 , - 7\right)$

Calculate the second derivative

$f ' ' \left(x\right) = 12 {x}^{2} - 24 x$

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

$12 {x}^{2} - 24 x = 0$

$\implies$, $12 x \left(x - 2\right) = 0$

$\implies$, $x = 0$ and $x = 2$

The inflection points are $\left(0 , 20\right)$ and $\left(2 , 4\right)$

Build a variation chart to determine the concavities

$\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}$$\left(0 , 2\right)$$\textcolor{w h i t e}{a a a a}$$\left(2 , + \infty\right)$

$\textcolor{w h i t e}{a a a a}$$\text{ sign f''(x)}$$\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}$$+$

$\textcolor{w h i t e}{a a a a}$$\text{ f(x)}$$\textcolor{w h i t e}{a 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}$$\cup$

graph{x^4-4x^3+20 [-32.73, 32.24, -5.85, 26.6]}