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Let f: \mathbf{R} \rightarrow \mathbf{R} be defined as

f(x)=x^{3}+x-5

If  g\left ( x \right ) is a function such that f(g(x))=x, \quad \forall\,\,^{\prime} x^{\prime} \in \mathbf{R},  then g^{\prime}(63) is equal to ______________ .

Option: 1

\frac{1}{49}


Option: 2

\frac{3}{49}


Option: 3

\frac{43}{49}


Option: 4

\frac{91}{49}


Answers (1)

best_answer

\mathrm{Here\: g(f(x))=x \; means \; f(x)\, and \,g(x)}\text{are inverse of each other.}

\mathrm{Now, g^{\prime}(f(x)) f^{\prime}(x)=1.}
\mathrm{\Rightarrow g^{\prime}(f(x))=\frac{1}{f^{\prime}(x)}----(1) }
\mathrm{\text { Now } f(x)=63 \Rightarrow x^{3}+x-5=63 .}
\mathrm{\Rightarrow x^{3}+x-68=0}
So \mathrm{x=4} Satisfies the above eqn

\mathrm{g^{\prime}(63) =\frac{1}{f^{\prime}(4)} \text { from eq }{ }^{n}-\text { (1) }}
                \mathrm{=\frac{1}{3(4)^{2}+1}=\frac{1}{49}}

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Rishi

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