Question

Let $f(x)=\underset{2}{\overset{x}{\int }}\frac{dt}{\sqrt{1+t^4}}$ and $g$ be the inverse of $f$. Then the value of $g^{\prime }(0)$ is -

• A. 1
• B. 17
• C. $\sqrt{17}$
• D. none of these

Answer 1

Answer: (C)

$f(x)=\underset{2}{\overset{x}{\int }}\frac{dt}{\sqrt{1+t^4}}[\because f(2)=0]$
$g(f(x))=x$
$g^{\prime }(f(x))f^{\prime }(x)=1$
$g^{\prime }(0)f^{\prime }(2)=2$
$g^{\prime }(0)=\frac{1}{f^{\prime }(2)}=\sqrt{17}$
$\Rightarrow f^{\prime }(x)=\frac{1}{\sqrt{1+x^4}}\Rightarrow f^{\prime }(2)=\frac{1}{\sqrt{17}}$