If the function f(x)=x^3+e^ (x)/(2) and g(x)=f^ -1 (x) , then the value of g^ (1) is

Correct answer: 0 Given, g\ f(x)\ =x array lc & g^ \ f(x)\ f^ (x)=1 If & f(x)=1 x=0, f(0)=1 array Substitute x=0 in Eq. (i), we get aligned & g^ (1)= 1 f^ (0) & g^ (1)=2 & [ ^ (x)=3 x²+(1)/(2)e^ x / 2 f^ (0)=(1)/(2)] & aligned

JEE Advanced 2009Paper 2

Exam: JEE Advanced
Year: 2009
Paper: Paper 2
Subject: Mathematics
Chapter: Differentiation

JEE Advanced 2009 · Paper 2 · Mathematics · Differentiation

If the function f(x)=x^3+e^ (x)/(2) and g(x)=f^ -1 (x) , then the...

If the function

f(x)=x^3+e^{\frac{x}{2}}

and

g(x)=f^{-1}(x)

, then the value of

g^{\prime}(1)

Correct answer

Step-by-step solution

Given,

g\{f(x)\}=x

\begin{array}{lc} \Rightarrow& g^{\prime}\{f(x)\} f^{\prime}(x)=1 \\ \text{ If }& f(x)=1 \Rightarrow x=0, f(0)=1 \end{array}

Substitute

x=0

in Eq. (i), we get

\begin{aligned}& g^{\prime}(1)=\frac{1}{f^{\prime}(0)}\\ & \Rightarrow \quad g^{\prime}(1)=2 \\ & {\left[\becausef^{\prime}(x)=3 x²+\frac{1}{2}e^{x / 2} \Rightarrow f^{\prime}(0)=\frac{1}{2}\right]} \\ & \end{aligned}

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