Chapter 2.3, Problem 62E

To determine

To show: The function $f\left(x\right)$ is either positive or negative in an interval $\left[a,b\right]$.

Explanation of Solution

Given information: The function $f\left(x\right)$ is continuous and never zero on interval $\left[a,b\right]$.

Proof:

As the function is continuous and never zero on interval $\left[a,b\right]$ . By the intermediate value theorem the function $f\left(x\right)$ takes every value between $f\left(a\right)$ and $f\left(b\right)$.

As $f\left(x\right)$ is never zero for $a<x<b$ . So, the function $f\left(x\right)$ never changes the sign in the interval $\left[a,b\right]$.

So, the function is either negative or positive for all the values of $x$ in $\left[a,b\right]$.

Hence, the function $f\left(x\right)$ is either positive or negative on an interval $\left[a,b\right]$.