Question

Let $f\left(x\right),g\left(x\right)$ be two continuously differentiable functions satisfying the relationships $f^{{}^{\prime }}\left(x\right)=g\left(x\right)$ and $f^{{}^{\prime \prime }}\left(x\right)=-f\left(x\right)$ . 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

Answer: 5

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
$h^{{}^{\prime }}\left(x\right)=2f\left(x\right)f^{{}^{\prime }}\left(x\right)+2g\left(x\right)g^{{}^{\prime }}\left(x\right)$
$=-2f\left(x\right)f^{{}^{\prime \prime }}\left(x\right)+2f\left(x\right)f^{{}^{\prime \prime }}\left(x\right)$
$\left(\therefore g^{(x)}=f^{\prime }(x),g^{\prime }(x)=f^{\prime \prime }(x)\right)$
$h^{{}^{\prime }}\left(x\right)=0$
$h\left(x\right)=$ constant
$\Rightarrow h\left(0\right)=5$
$h\left(10\right)=5$ .