Find the coordinates of the points of inflection for the function.
-$f(x)=\frac{5 x}{x^{2}+49}$
A) $\left( \pm \frac{1}{\sqrt{3}}, \frac{21}{4}\right)$
B) $(0,0),\left(\sqrt{147},-\frac{5}{196} \sqrt{147}\right)$,
C) $\left( \pm \frac{7}{\sqrt{3}}, \frac{15}{196}\right)$
$\left(-\sqrt{147}, \frac{5}{196} \sqrt{147}\right)$
D) $(0,0),\left(\sqrt{147}, \frac{5}{196} \sqrt{147}\right)$, $\left(-\sqrt{147},-\frac{5}{196} \sqrt{147}\right)$

Question

Quizplus · Accepted Answer

To find the points of inflection, we need to find the second derivative of the function and set it equal to zero. The second derivative is:$$f''(x) = \frac{544x^2 - 4900}{(x^2 + 49)^3}$$Setting this equal to zero and solving for x, we get:$$x = \pm \sqrt{147}$$Substituting these values back into the original function, we get the points of inflection:$$(0,0), \left(\sqrt{147}, \frac{5}{196} \sqrt{147}\right), \left(-\sqrt{147},-\frac{5}{196} \sqrt{147}\right)$$

Find the Coordinates of the Points of Inflection for the Function