Question

Let $f$ and $g$ be two differentiable functions such that $f^{\prime }\left(x\right)=\varphi \left(x\right)$ and ${\varphi }^{\prime }\left(x\right)=f\left(x\right)$ for all real $x.$ If $f\left(3\right)=5$ and $f^{\prime }\left(3\right)=4,$ then the value of $(f\left(10\right){)}^2-(\varphi \left(10\right){)}^2$ is

Answer 1

Let $h\left(x\right)=(f\left(x\right){)}^2-(\varphi \left(x\right){)}^2$
$\Rightarrow h^{\prime }\left(x\right)=2f\left(x\right)\cdot f^{\prime }\left(x\right)-2\varphi \left(x\right)\cdot {\varphi }^{\prime }\left(x\right)$
$\Rightarrow h^{\prime }\left(x\right)=2f\left(x\right)\cdot f^{\prime }\left(x\right)-2f^{\prime }\left(x\right)\cdot f\left(x\right)$
$(\because f^{\prime }\left(x\right)=\varphi \left(x\right),{\varphi }^{\prime }\left(x\right)=f\left(x\right))$
$\Rightarrow h^{\prime }\left(x\right)=0\forall x\in R$
$\Rightarrow h\left(x\right)$ is a constant function.
$\Rightarrow h\left(10\right)=h\left(3\right)$
$=(f\left(3\right){)}^2-(\varphi \left(3\right){)}^2$
$=(f\left(3\right){)}^2-(f^{\prime }\left(3\right){)}^2$
$=5^2-4^2=9$