Prove that $f(x)=x^2$ is concave up.

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Lesson Parent: 
Geometry of Functions IX: Concavity
Problem: 

Prove that $f(x)=x^2$ is concave up.

Answer: 

The function $x^2$ is concave up.

SAMPLE SOLUTION

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Solution: 

Recall that a function is concave up on an open interval when its derivative is increasing. So to prove that $x^2$ is concave up we need to prove that when we go from left to right that the slope is always increasing.

First, let's pick two arbitrary values $x_1$ and $x_2$ and assume that <Sign in to see all the formulas>. Second, we know that the derivative of $x^2$ is $2x$. With our usual notation we have <Sign in to see all the formulas>.

Now for some simple algebra:

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Since, <Sign in to see all the formulas> means that <Sign in to see all the formulas> is increasing, <Sign in to see all the formulas> is concave up.