Find the critical points of $f(x)=x^3$.

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Lesson Parent: 
Geometry of Functions VI: Critical Points
Problem: 

Find the critical points of $f(x)=x^3$.

Answer: 

$x=0$ is the only critical point of $f(x)=x^3$.

SAMPLE SOLUTION

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Solution: 

Critical points are where either <Sign in to see all the formulas> or <Sign in to see all the formulas> does not exist. We know that the derivative is <Sign in to see all the formulas> exists and is <Sign in to see all the formulas>. We now set <Sign in to see all the formulas> and easily see that

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Note that while $x=0$ is a critical point of <Sign in to see all the formulas>, it is not a maximum or minimum.