Find the point(s) of inflection of the graph of $f(x)=\cos ^{-1} 7 x$
A) $(1,0)$ , $\left(0, \frac{\pi}{2}\right)$ , and $\left(0, \frac{\pi}{7}\right)$
B) $(1,0)$
C) $\left(0, \frac{\pi}{7}\right)$
D) $\left(0, \frac{\pi}{2}\right)$

Question

Quizplus · Accepted Answer

Using the chain rule, we can find the second derivative of $f(x)$ as follows:
\begin{align*}
f(x)&=\cos ^{-1} (7 x)\
f'(x)&=-\frac{7}{\sqrt{1-(7x)^2}}\
f''(x)&=\frac{49x}{(1-(7x)^2)^{3/2}}
\end{align*}
The second derivative is defined for all values of $x$, so there are no vertical asymptotes. It remains to find the zeros of $f''(x)$ to determine the points of inflection. Setting $f''(x)=0$ gives
$$x=0$$
since $f''(x)$ is never zero for $x\neq 0$. Therefore, the point of inflection is $\left(0, \frac{\pi}{2}\right)$. Thus, the correct answer is D.

Find the Point(s) of Inflection of the Graph Of f(x)=cos⁡−17xf(x)=\cos ^{-1} 7 xf(x)=cos−17x

Find the Point(s) of Inflection of the Graph Of $f(x)=\cos ^{-1} 7 x$