# How do you find all critical point and determine the min, max and inflection given V(w)=w^5-28?

May 7, 2018

Use Your First and Second Derivative

#### Explanation:

I first like to start with taking my first and second derivatives:
$V ' \left(w\right) = 5 {w}^{4}$
$V ' ' \left(w\right) = 20 {w}^{3}$

The min and max are points where $V ' \left(w\right) = 0$ so lets start with them.

$0 = 5 {w}^{4}$
$0 = {w}^{4}$
$0 = w$

Now we test to see if $w = 0$ is a min, max, or none. We do this by using the second derivative. If our Final answer is greater than zero we know it's a minimum, if our final answer is less than zero we know it's a maximum and if our final answer is zero we know it's a turning point (think x^3 @ x=0).

$V ' ' \left(0\right) = 20 {\left(0\right)}^{3}$
$V ' ' \left(0\right) = 0$

Since our answer is zero we know its a turning point.

This tells us that this equations has no Max or a Min. However, it does have a turning point at $w = 0$