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

Explanation:

Step1: Recall critical - point definition

Critical points occur where $f^{\prime}(x)=0$ or $f^{\prime}(x)$ is undefined. The derivative $f^{\prime}(x)=\frac{-x^{2}+9}{(x^{2}+9)^{2}}$ is defined for all real $x$ since the denominator $(x^{2}+9)^{2}>0$ for all $x\in R$ (because $x^{2}\geq0$ for all $x\in R$, so $x^{2}+9 > 0$).

Step2: Set the derivative equal to zero

Set $f^{\prime}(x) = 0$, so $\frac{-x^{2}+9}{(x^{2}+9)^{2}}=0$. Since the denominator $(x^{2}+9)^{2}\neq0$ for all real $x$, we only need to solve the numerator equal to zero. Set $-x^{2}+9 = 0$. Rearrange to get $x^{2}=9$. Take the square - root of both sides: $x=\pm3$.

Answer:

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