the maximum amount of $m\sin{kx} + n\cos{px}$

Question

for all real numbers x, find the maximum amount of $$m\sin{kx} + n\cos{px}$$ I know that we could solve this problem by using differential calculus: $$ f(x) = m\sin{kx} + n\cos{px} \therefore f^\prime(x) = mk\cos(kx) - np\sin(px) $$ the we have: $$ \frac{mk}{np}\cos(kx) = \sin(px) \therefore x = \frac{1}{p}\arcsin(\frac{mk}{np}\cos(kx))$$ but this is not what I am looking for and I don't know how to solve it. please help.