Question

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

• A. $10$
• B. $5$
• C. $15$
• D. None of these

Answer 1

Answer: (B)

Since $f’(x)=g(x),f^{\prime }(x)=g^{\prime }(x)$
Put $f^{\prime }(x)=-f(x)$. Hence $g^{\prime }(x)=-f(x)$
we have $h^{\prime }(x)=2f(x)f’(x)+2g(x)g^{\prime }(x)$
$=2[f(x)g(x)+g(x)[-f(x)]]=2[f(x)g(x)-f(x)g(x)]=0$
$\therefore h(x)=C$, a constant
$\therefore h(0)=Ci.e.C=5$
$h(x)=5$ for all $x$. Hence $h(10)=5.$