Prove that $f(x)=x^2$ decreases on the interval $(-\infty,0]$ and increases on the interval $[0,\infty)$.

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Lesson Parent: 
Geometry of Functions I: Increasing and Decreasing Functions
Problem: 

Prove that $f(x)=x^2$ decreases on the interval $(-\infty,0]$ and increases on the interval $[0,\infty)$.

Answer: 

It is true that $f(x)=x^2$ decreases on the interval $(-\infty,0]$ and increases on the interval $[0,\infty)$.

SAMPLE SOLUTION

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Solution: 

Recall that our definitions of increasing and decreasing functions can be found here.

Let's look at the interval <Sign in to see all the formulas> where we want to prove $x^2$ is increasing. Let's arbitrarily choose two numbers <Sign in to see all the formulas> and <Sign in to see all the formulas> such that <Sign in to see all the formulas>. First, let's multiply both sides of <Sign in to see all the formulas> by $x_1$ to find that

<Sign in to see all the formulas>

Second, let's multiply both sides of <Sign in to see all the formulas> by $x_2$ to find that

<Sign in to see all the formulas>

Now putting the two inequalities together we find that

<Sign in to see all the formulas>

So now we have shown that in the interval <Sign in to see all the formulas> that when <Sign in to see all the formulas> then <Sign in to see all the formulas>. This is precisely the definition of an increasing function. Therefore, in the interval <Sign in to see all the formulas> <Sign in to see all the formulas> is an increasing function.

Let's look at the interval <Sign in to see all the formulas> where we want to prove $x^2$ is decreasing. Let's arbitrarily choose two numbers <Sign in to see all the formulas> and <Sign in to see all the formulas> such that <Sign in to see all the formulas>. First, let's multiply both sides of <Sign in to see all the formulas> by $x_1$ to find that

<Sign in to see all the formulas>

Note that the $\lt$ symbol has changed to a $\gt$ symbol because $x_1$ is negative. Second, let's multiply both sides of <Sign in to see all the formulas> by $x_2$ to find that

<Sign in to see all the formulas>

Note that the $\lt$ symbol has changed to a $\geq$ symbol because $x_2$ is either zero or negative.Now putting the two inequalities together we find that

<Sign in to see all the formulas>

So now we have shown that in the interval <Sign in to see all the formulas> that <Sign in to see all the formulas> when <Sign in to see all the formulas>. This is precisely the definition of a decreasing function. Therefore, in the interval <Sign in to see all the formulas> <Sign in to see all the formulas> is a decreasing function.