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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:

Problem:

Answer:

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

Solution:

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

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

$x_2$ to find that

Now putting the two inequalities together we find that

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.

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

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

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.