Chapter 10.5, Problem 8QR

To determine

To find: the given function is both positive and decreasing on some interval.

Answer to Problem 8QR

Hence, there is no interval such that $(N,\infty )$ where $f(x)$ is both positive and decreasing.

Explanation of Solution

Given information:

Given function is: $f(x)=\frac{3+x^2}{3-x^2}$ , whether function is both positive and decreasing on some interval $\left(N,\infty \right)$ .

Calculation:

The given function is, $f(x)=\frac{3+x^2}{3-x^2}$

We $f(x)=\frac{3+x^2}{3-x^2}>0,$ if $3-x^2>0$ That is $x\in (-\sqrt{3},\sqrt{3})$

So, $f(x)$ is positive only when $x\in (-\sqrt{3},\sqrt{3}),$ but that is not the interval we are looking for. Therefore, there is no interval such that $(N,\infty )$ where $f(x)$ is both positive and decreasing.

Hence, there is no interval such that $(N,\infty )$ where $f(x)$ is both positive and decreasing.