Chapter 3.1, Problem 6AYU

True or False The graph of the function $f\left(x\right)=x^2$ is increasing on the interval $[0,\infty )$. (p. 84)

To determine

Whether the given statement is true or false.

Answer to Problem 6AYU

Solution:

True.

Given:

The given statement is

“The graph of the function $f(x)=x^2$ is increasing on the interval $(\text{0},\infty )$.”

Explanation of Solution

A function is said to be increasing in an interval $I$, if for any 2 points $a$ and $b$ in $I$, with $a<b$, we have $f(a)\text{ < }f(b)$.

Here, let the 2 points be $a = 1$ and $b = 2$.

Therefore, we have $f(1) = 1^2 = 1$ and $f(2) = 2^2 = 4$.

Thus, we get $f(1) < f(2)$, where $1 < 2$.

Similarly, we can see that $f(a) < f(b)$ is always true for any 2 points, such that $a < b$, in the given interval.

Therefore, the given statement is true.