Question

If $f\left(x\right)$ is a twice differentiable function such that $f^{"}\left(x\right)=-f\left(x\right), \, f^{'}\left(x\right)=g\left(x\right),$ $h\left(x\right)=f^{2}\left(x\right)+g^{2}\left(x\right)$ and $h\left(10\right)=10,$ then $h\left(5\right)$ is equal to

• A. $5$
• B. $15$
• C. $10$
• D. $17$

Answer 1

Answer: (C)

$h^{'}\left(x\right)=2f\cdot f^{'}+2gg^{'}$
$=2f\cdot f^{'}+2\left(f^{'}\right)\left(f^{"}\right)\left(\right.\because g=f^{'}\left.\right)$
$=2f\cdot f^{'}+2f^{'}\left(- f\right)=0$
$\therefore h^{'}\left(x\right)=0\Rightarrow h\left(x\right)$ is a constant function
$\therefore h\left(x\right)=10\Rightarrow h\left(5\right)=10$