Q110 / 180 · COMEDK 2016 · Mathematics
If $f(x)$ is a function such that $f^{\prime \prime}(x)+f(x)=0$ and $g(x)=[f(x)]^{2}+\left[f^{\prime}(x)\right]^{2}$ and $g(3)+8$, then $g(8)=$
- Correct answerA8
- B3
- C0
- D5
Solution
We have,
$g(x)=[f(x)]^{2}+\left[f^{\prime}(x)\right]^{2}$
Differentiate the function $g(x)$
$g^{\prime}(x)=2 f(x) f^{\prime}(x)+2 f^{\prime}(x) f^{\prime \prime}(x)$
Use chain rule,
$2 f^{\prime}(x)\left[f(x)+f^{\prime \prime}(x)\right]=2 f^{\prime}(x)(0)=0$
Hence, $g(x)$ is $a$ constant function $\Rightarrow \quad g(x)=c$, constant
But, $g(3)=8$, so $g(x)=8$
For all real $x$.
Hence, $g(8)=8$
Hence, the correct answer is option: 'A'.