Question 94
Consider the function $f(x) = a(|\sin x| + |\cos x|) - 3\sin 2x - 7$, where $a$ is a real parameter.
a) Prove that $f(x) = f(\frac{\pi}{2} - x) = f(\pi + x) = f(\frac{3\pi}{2} - x)$ for every $x$.
b) Find all pairs $(a, n)$, where $n$ is a positive integer, for which the equation $f(x) = 0$ has $2007$ roots in the interval $(0, n\pi)$.
a) Prove that $f(x) = f(\frac{\pi}{2} - x) = f(\pi + x) = f(\frac{3\pi}{2} - x)$ for every $x$.
b) Find all pairs $(a, n)$, where $n$ is a positive integer, for which the equation $f(x) = 0$ has $2007$ roots in the interval $(0, n\pi)$.