I have no idea how to find the maximum of the following expression,
$a(t)b(t)+c(t)d(t)$
where the functions $b$ and $d$ satisfy $b(t)^2+d(t)^2=1$
Anyone have a clue?
$t$ is in some compact intervall and the functions $a,c$ are never zero and differentiable on it.
Given that $b(t)^2 + d(t)^2 = 1$, you can always find a substution such that $b(t) = \cos(\theta)$ and $d(t) = \sin(\theta)$ with $\theta(t) = \tan^{-1} \left( \frac{d(t)}{b(t)} \right) $
This gives the following result
$$ \frac{{\rm d}}{{\rm d}t} (a(t) b(t)+c(t) d(t)) = b(t) ( a'(t)+c(t) \dot{\theta}) + d(t) ( c'(t)-a(t) \dot{\theta}) = 0 $$
Which is solved when
$$ \dot{\theta} = \frac{ a'(t) b(t)+c'(t) d(t)}{a(t) d(t)-c(t) b(t)} $$
From there without further knowledge of the functions, we cannot proceed.