Question

The functions $f , g$ and $h$ satisfy the relations $f ^{'}\left(x\right)=g\left(x+1\right)$. Then $f ^{"}\left(2x\right)$ is equal to

• A. $h\left(2x\right)$
• B. $4h\left(2x\right)$
• C. $h\left(2x-1\right)$
• D. $h\left(2x+1\right)$

Answer 1

Answer: (A)

$h\left(2x\right)$

Answer 2

We have,
$f^{\prime}(x)= g(x+1)$
$\Rightarrow f^{\prime \prime}(x) =g^{\prime}(x+1)$
$\text { But } g^{\prime}(x)=h(x-1)$
$\Rightarrow g^{\prime}(x+1) =h(x+1-1)$
$=h(x)$
$\therefore f^{\prime \prime}(x)=h(x)$
$\Rightarrow f^{\prime \prime}(2 x) =h(2 x)$