Question

If $f$ and $g$ be differentiable function satisfying $g^{\prime }\left(a\right)=2,g\left(a\right)=b$ and $fog=I$ (identity function), then $f^{\prime }\left(b\right)$ is equal to

• A. $\frac{1}{2}$
• B. $2$
• C. $\frac{2}{3}$
• D. None of these

Answer 1

The correct option is A $\frac{1}{2}$

Since, $fog=I$
$\Rightarrow fog\left(x\right)=x$ for all $x$

$\Rightarrow f^{\prime }\left\{g\right(x\left)\right\}\cdot g^{\prime }\left(x\right)=1$ for all $x$

$\Rightarrow f^{\prime }\left\{g\right(a\left)\right\}\cdot g^{\prime }\left(a\right)=1$
$\Rightarrow f^{\prime }\left\{g\right(a\left)\right\}=\frac{1}{g^{\prime }\left(a\right)}=\frac{1}{2}$
$\Rightarrow f^{\prime }\left(b\right)=\frac{1}{2}$