Question

If $f(x)$ is a function such that $f'(x) +f(x) = 0$ and $g(x) =[f(x)]^2+ [f'(x)]^2 $ and $g(3) = 8$ ,then $g(8)=$

• A. 8
• B. 3
• C. 0
• D. 5

Answer 1

Answer: (A)

We have, $g(x)=[f(x)]^{2}+\left[f^{\prime}(x)\right]^{2}$
Differentiate the function $g ( x )$
$\Rightarrow g ^{\prime}( x )=2 f ( x ) f ^{\prime}( x )+2 f ^{\prime}( x ) f ^{\prime \prime}( x )$, use chain rule
$=2 f ^{\prime}( x )\left[ f ( x )+ f ^{\prime \prime}( x )\right]=2 f ^{\prime}( x )(0)=0$, use the given condition
Hence $g ( x )$ is a constant function
$\Rightarrow g ( x )= c$, constant
But $g (3)=8, \operatorname{so} g ( x )=8$, for all real $x$
Hence $g(8)=8$