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 $f^n\left(x\right)=-f\left(x\right)$. Let $h\left(x\right)=\left[f\right(x){]}^2+[g\left(x\right){]}^2$. If $h\left(0\right)=5$, then $h\left(10\right)=$

• A. $10$
• B. $5$
• C. $15$
• D. $20$

Answer 1

Answer: (B)

$5$

Given:

$g\left(x\right)=f^{\prime }\left(x\right)$ ...... $\left(1\right)$

$\therefore g^{\prime }\left(x\right)=f^{\prime \prime }\left(x\right)=-f\left(x\right)$ ...[2] $[\because f^n(x)=-f(x\left)\right]$

$h\left(x\right)=\left[f\right(x){]}^2+[g\left(x\right){]}^2$

$\therefore h^{\prime }\left(x\right)=2f\left(x\right)f^{\prime }\left(x\right)+2g\left(x\right)g{}^{\prime }\left(x\right)$

$\therefore h^{\prime }\left(x\right)=2f\left(x\right)f{}^{\prime }\left(x\right)+2f{}^{\prime }\left(x\right)(-f(x\left)\right)$ ...(using [1] and [2])

$\therefore h^{\prime }\left(x\right)=0$

$\therefore h\left(x\right)$ is a constant function

$\therefore h\left(10\right)=h\left(0\right)=5$