Question

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

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

Answer 1

The correct option is B $5$

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

And $f^{{}^{\prime \prime }}\left(x\right)=-f\left(x\right)$

Differentiate 1st equation ;

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

$\Rightarrow -f\left(x\right)=g^{{}^{\prime }}\left(x\right)$

$\Rightarrow g^{{}^{\prime }}\left(x\right)=-f\left(x\right)$ $riightarrow$equation (1)

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

Differentiate on both sides ;

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

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

$\Rightarrow h^{{}^{\prime }}x=0$

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

Let $h\left(x\right)=k$ (some constant 'k')

Given $h\left(0\right)=5$

$\Rightarrow k=5$

$\Rightarrow h\left(x\right)=k=5$

$\Rightarrow h\left(10\right)=5$