Question

find the critical points of the following function.

f(x)=\frac{x}{x^{2}+49}

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. the critical point(s) occur(s) at x=
(use a comma to separate answers as needed.)
b. there are no critical points.

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 \(u = x\), \(u^\prime=1\); \(v=x^{2}+49\), \(v^\prime = 2x\).

$$
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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 \(49 - x^{2}=0\).

$$
LATEXBLOCK1
$$

Answer:

A. The critical point(s) occur(s) at \(x = - 7,7\)