find the critical points of the following function. f(x)=x/(x^2 + 9) what is the derivative of f(x)=x/(x^2 +…

find the critical points of the following function. f(x)=x/(x^2 + 9) what is the derivative of f(x)=x/(x^2 + 9)? f(x)=(-x^2 + 9)/(x^2 + 9)^2 select the correct choice below and, if necessary, fill in the answer box to complete a. the critical point(s) occur(s) at x= (use a comma to separate answers as needed.) b. there are no critical points.

find the critical points of the following function. f(x)=x/(x^2 + 9) what is the derivative of f(x)=x/(x^2 + 9)? f(x)=(-x^2 + 9)/(x^2 + 9)^2 select the correct choice below and, if necessary, fill in the answer box to complete a. the critical point(s) occur(s) at x= (use a comma to separate answers as needed.) b. there are no critical points.

Answer

Explanation:

Step1: Recall critical - point definition

Critical points occur where $f^{\prime}(x)=0$ or $f^{\prime}(x)$ is undefined. The denominator of $f^{\prime}(x)=\frac{-x^{2}+9}{(x^{2}+9)^{2}}$ is $(x^{2}+9)^{2}$. Since $x^{2}\geq0$ for all real $x$, then $x^{2}+9>0$ and $(x^{2}+9)^{2}>0$ for all real $x$. So, we only need to find where $f^{\prime}(x) = 0$.

Step2: Set the numerator equal to zero

Set $-x^{2}+9 = 0$. Rearrange the equation to get $x^{2}=9$.

Step3: Solve for x

Taking the square - root of both sides, we have $x=\pm3$.

Answer:

A. The critical point(s) occur(s) at $x=-3,3$