Let $f$ be a real-valued function defined on the interval $\left( { - 1,\,1} \right)$ such that ${e^{ - x}}\,\,f\,\,\left( x \right) = 2\,\, + \,\,\mathop \smallint \nolimits_0^x \,\,\sqrt {{t^4}\,\, + \,\,1} \,\,dt$, for all $x \in \left( { - 1,\,\,1} \right)$ and let ${f^{ - 1}}$ be the inverse function of $f$. Then, ${\left( {{f^{ - 1}}} \right)^\prime }\,\,\left( 2 \right)$ is equal to

JEE MATHEMATICS MCQ: INDEFINITE INTEGRATION

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Application +4 | -1

Let

f

be a real-valued function defined on the interval

\left( { - 1,\,1} \right)

such that

{e^{ - x}}\,\,f\,\,\left( x \right) = 2\,\, + \,\,\int \nolimits_0^x \,\,\sqrt {{t^4}\,\, + \,\,1} \,\,dt

, for all

x \in \left( { - 1,\,\,1} \right)

and let

{f^{ - 1}}

be the inverse function of

f

. Then,

{\left( {{f^{ - 1}}} \right)^\prime }\,\,\left( 2 \right)

is equal to

View answer and explanation

Correct option: B

\frac{1}{3}

Option B (1/3) is correct because after differentiating and using the inverse function property, the derivative of the inverse function at 2 is found to be 1/3.

JEE MATHEMATICS MCQ: INDEFINITE INTEGRATION

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