Question

The point of inflection for the function $f\left(x\right)={\sin }^{-1}x$ is:

• A. $x=\frac{1}{2}$
• B. $x=\frac{1}{\sqrt{2}}$
• C. $x=0$
• D. $x=-\frac{1}{2}$

Answer 1

The correct option is C $x=0$

Given: $f\left(x\right)={\sin }^{-1}x$
$\Rightarrow f^{\prime }\left(x\right)=\frac{1}{\sqrt{1-x^2}}=(1-x^2{)}^{-1/2}$
$\Rightarrow f^{\prime \prime }\left(x\right)=-2x\left(\frac{-1}{2}\right)(1-x^2{)}^{-3/2}$
$\Rightarrow f^{\prime \prime }\left(x\right)=\frac{x}{(1-x^2{)}^{\frac{3}{2}}}$
Clearly, the second derivative is zero at $x=0$ and also it changes sign at $x=0$. Hence it is a point of inflection.