find the critical points of the following function.\n\n( f(x)=\frac{x}{x^{2}+25} )\n\nselect the correct…

find the critical points of the following function.\n\n( f(x)=\frac{x}{x^{2}+25} )\n\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\na. the critical point(s) occur(s) at ( x= ) (use a comma to separate answers as needed.)\nb. there are no critical points.

find the critical points of the following function.\n\n( f(x)=\frac{x}{x^{2}+25} )\n\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\na. the critical point(s) occur(s) at ( x= ) (use a comma to separate answers as needed.)\nb. there are no critical points.

Answer

Explanation:

Step1: Find the derivative of (f(x))

Use the quotient rule (\left(\frac{u}{v}\right)'=\frac{u'v - uv'}{v^{2}}). Here (u = x), (u'=1), (v=x^{2}+25), (v' = 2x). [ \begin{align*} f'(x)&=\frac{(1)(x^{2}+25)-x(2x)}{(x^{2}+25)^{2}}\ &=\frac{x^{2}+25 - 2x^{2}}{(x^{2}+25)^{2}}\ &=\frac{25 - x^{2}}{(x^{2}+25)^{2}} \end{align*} ]

Step2: Set (f'(x)=0)

Set the numerator equal to zero (since the denominator ((x^{2}+25)^{2}>0) for all real (x)). [ \begin{align*} 25 - x^{2}&=0\ x^{2}&=25\ x&=\pm5 \end{align*} ]

Answer:

A. The critical point(s) occur(s) at (x=- 5,5)