Prove that f(x)=x2 is concave up.

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Lesson Parent: 
Geometry of Functions IX: Concavity
Problem: 

Prove that f(x)=x2 is concave up.

Answer: 

The function x2 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 x2 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 x1 and x2 and assume that <Sign in to see all the formulas>. Second, we know that the derivative of x2 is 2x. With our usual notation we have <Sign in to see all the formulas>.

Now for some simple algebra:

<Sign in to see all the formulas>

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.