IIT-JEE 2010 · Paper 2 · Mathematics Q22

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MathematicsSingle correct+5 / -2Question 22 of 57

Let $f$ be a real-valued function defined on the interval $(-1,1)$ such that $e^{-x}f(x)=2+\displaystyle\int_0^x\sqrt{t^4+1}\,dt$ , for all $x\in(-1,1)$ , and let $f^{-1}$ be the inverse function of $f$ . Then $(f^{-1})'(2)$ is equal to

\dfrac13

\dfrac12

\dfrac1e

Answer and worked solution

Official answer: B

Find $f$ and $f'$ at 0. $f(x)=e^x\left(2+\int_0^x\sqrt{t^4+1}\,dt\right)$ , so $f(0)=2$ and
$f'(x)=e^x\left(2+\int_0^x\sqrt{t^4+1}\,dt\right)+e^x\sqrt{x^4+1},\qquad f'(0)=2+1=3.$

Inverse. $f^{-1}(2)=0$ , so $(f^{-1})'(2)=\dfrac1{f'(0)}=\dfrac13$ .

(B)

Topics:Derivatives of standard functions Fundamental theorem of calculus Composition and inverse functions

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IIT-JEE 2010 · Paper 2, Mathematics question 22