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

First, we compute the first and second derivatives of $f(x)$: \begin{align*} f(x) &= \cos^{-1}(7x) \ f'(x) &= -\frac{7}{\sqrt{1-49x^2}} \ f''(x) &= \frac{147x}{(1-49x^2)^{3/2}} \end{align*} The second derivative is 0 exactly when $x=0$. So, we just need to check if there is a change in concavity at $x=0$. We observe that $f''(x)$ changes sign at $x=0$ since $f''(x)>0$ for $x<0$ and $f''(x)<0$ for $x>0$. Therefore, there is a point of inflection at $x=0$ and the 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 $f ( x ) = \cos ^ { - 1 } 7 x$