# If f''(x)= -f(x) and g(x)= f'(x) and F(x) = (f(x/2))^2 + (g(x/2))^2 and given that F(5) = 5 then F(10) equals ? (hint : answer is an integer from 0 - 9)

May 14, 2018

5

#### Explanation:

Let $G \left(x\right) \equiv {\left(f \left(x\right)\right)}^{2} + {\left(g \left(x\right)\right)}^{2}$. Then

$\frac{\mathrm{dG}}{\mathrm{dx}} = 2 f \left(x\right) {f}^{'} \left(x\right) + 2 g \left(x\right) {g}^{'} \left(x\right)$
$q \quad = - 2 f ' ' \left(x\right) g \left(x\right) + 2 g \left(x\right) f ' ' \left(x\right) = 0$

where we have used $f ' ' \left(x\right) = - f \left(x\right)$, ${f}^{'} \left(x\right) = g \left(x\right)$and ${g}^{'} \left(x\right) = f ' ' \left(x\right)$

Thus $G \left(x\right)$ is a constant, and so is $F \left(x\right) = G \left(\frac{x}{2}\right)$.

Since $F \left(5\right) = 5$, we must have $F \left(10\right) = 5$