Question

Let $f: R \rightarrow R$ be a function defined as $f(x)=x^3+x-1$ and $g$ is inverse of $f$, then the value of $\frac{ g (9)}{ g ^{\prime}(9)}$ is equal to

• A. 10
• B. 15
• C. 20
• D. 26

Answer 1

Answer: (D)

$ \because g ^{\prime}( f ( x ))=\frac{1}{ f ^{\prime}( x )}$
Here $f(x)=9$, when $x=2$
$\therefore g ^{\prime}(9)=\frac{1}{ f ^{\prime}(2)}$
Clearly $f^{\prime}(x)=3 x^2+1 \Rightarrow f^{\prime}(2)=13$

$\therefore g ^{\prime}(9)=\frac{1}{13} $
$\because f (2)=9 \Rightarrow f ^{-1}(9)= g (9)=2$
$\therefore \frac{ g (9)}{ g ^{\prime}(9)}=26$