find the critical points of the following function.\n\nf(x)=\\frac{x}{x^{2}+49}\n\nselect the correct choice…
find the critical points of the following function.\n\nf(x)=\\frac{x}{x^{2}+49}\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=\n(use a comma to separate answers as needed.)\nb. there are no critical points.
Answer
Explanation:
Step1: Find the derivative of the function
Use the quotient rule ((\frac{u}{v})^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}). Here (u = x), (u^\prime=1); (v=x^{2}+49), (v^\prime = 2x). [ \begin{align*} f^\prime(x)&=\frac{1\times(x^{2}+49)-x\times(2x)}{(x^{2}+49)^{2}}\ &=\frac{x^{2}+49 - 2x^{2}}{(x^{2}+49)^{2}}\ &=\frac{49 - x^{2}}{(x^{2}+49)^{2}} \end{align*} ]
Step2: Set the derivative equal to zero
Set (f^\prime(x)=0), so (\frac{49 - x^{2}}{(x^{2}+49)^{2}}=0). Since the denominator ((x^{2}+49)^{2}>0) for all real (x), we solve (49 - x^{2}=0). [ \begin{align*} 49 - x^{2}&=0\ x^{2}&=49\ x&=\pm7 \end{align*} ]
Answer:
A. The critical point(s) occur(s) at (x = - 7,7)