Question

Let $f\left(x\right),g\left(x\right)$ be two continuously differentiable functions satisfying the relationships $f^{'}\left(x\right)=g\left(x\right)$ and $f^{''}\left(x\right)=-f\left(x\right)$ . Let $h\left(x\right)=f \left(x\right)^{2}+g \left(x\right)^{2}$ . If $h\left(0\right)=5$ . then find value of $h\left(10\right)$ .

Answer 1

Given that
$h\left(x\right)=\left(f \left(x\right)\right)^{2}+\left(g \left(x\right)\right)^{2}$
Differentiate on both sides with respect to $x$ on both sides
$h^{'}\left(x\right)=2f\left(x\right)f^{'}\left(x\right)+2g\left(x\right)g^{'}\left(x\right)$
$=-2f\left(x\right)f^{''}\left(x\right)+2f\left(x\right)f^{''}\left(x\right)$
$\left(\therefore g^{(x)}=f^{\prime}(x), g^{\prime}(x)=f^{\prime \prime}(x)\right)$
$h^{'}\left(x\right)=0$
$h\left(x\right)=$ constant
$\Rightarrow h\left(0\right)=5$
$h\left(10\right)=5\,$ .