Limit Continuity And Differentiability Ques 28
- Let $f$ be a twice differentiable function such that
$f^{\prime \prime}(x)=-f(x), f^{\prime}(x)=g(x)$ and
$ h(x)=[f(x)]^2+[g(x)]^2 $
Find $h(10)$, if $h(5)=11$.
(1983, 3M)
Show Answer
Solution: Given, $h(x)=[f(x)]^2+[g(x)]^2$
$ \begin{aligned} & \Rightarrow \quad h^{\prime} x=2 f(x) \cdot f^{\prime}(x)+2 g(x) \cdot g^{\prime}(x) \\ & =\quad 2[f(x) \cdot g(x)-g(x) \cdot f(x)] \\ & =\quad 0 \quad\left[\because f^{\prime}(x)=g(x) \text { and } g^{\prime}(x)=-f(x)\right] \\ \end{aligned} $
$\therefore \quad h(x)$ is constant.
$ \Rightarrow \quad h(10)=h(5)=11 $
BETA
