Where on the x-axis is $f(x)=x^4-4x^3$ increasing or decreasing?

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Lesson Parent: 
Geometry of Functions IV: Using Derivatives To Identify Increasing and Decreasing Functions
Problem: 

Where on the x-axis is $f(x)=x^4-4x^3$ increasing or decreasing?

Answer: 

$f(x)=x^4-4x^3$ is decreasing when $x\lt 3$ and is increasing when $x\gt 3$.

SAMPLE SOLUTION

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Solution: 

First, let's graph the function to see where we think it might be increasing or decreasing.

Graph of f(x)=x^4-4x^3

It appears to be decreasing somewhere left of $x=0$ and increasing somewhere to the right. Let's use the function's derivative to see if this is correct.

This function is a polynomial and we know the derivative of a polynomial. Therefore,

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Because $4x^2$ is always greater than zero we know that <Sign in to see all the formulas> when <Sign in to see all the formulas> and <Sign in to see all the formulas> when <Sign in to see all the formulas>. Therefore, $f(x)$ is decreasing when <Sign in to see all the formulas> and is increasing when <Sign in to see all the formulas>.

Our visual guess of decreasing to the left of zero and increasing to the right of zero was incorrect. We could have spent a lot of time graphing the function trying to find exactly where this point was, but using the derivative to find it was easier.