Applications of differentiation

Question

Let $f$ be a twice differentiable function such that $f ^{\prime\prime}(x)=-f(x)$ and $f'(x)=g(x)$ for all $x \in \mathbb{R}$. If $h(x) =[f(x)]^2+[g(x)]^2$ for all $x \in \mathbb{R}$, then find $h(10)$ if $h(5)=11$. How can do this sum without using differential equation? Must be done using techniques restricted to Calculus 1.

Answer 1

Consider the derivative of $h(x) = (f(x))^2 + (g(x))^2$

$$h'(x) = 2(f(x)f'(x) + g(x)g'(x))$$

Note that $g'(x) = f^{\prime \prime}(x)= -f(x)$

The above makes $h'(x)= 0$

Using the fact that $h(5)= 11$, you should be able to finish off

Answer 2

We have that $f''(x)=g'(x)=-f(x)$ and $f'(x)=g(x)$. Then $$h'(x)=2f(x)f'(x)+2g(x)g'(x)$$ $$=2f(x)g(x)-2g(x)f(x)=0$$

Thus $$h'(x)=0$$

Can you end it?