This is a contest math question that I don't remember the reference.
When does $f(x;\alpha)=\cos(\alpha x)-\sin^2x-1$ has unique zero?
Obviously, $f(0;\alpha)=0$ for all $\alpha\in{\mathbb R}$. I've no idea what theorem can be used here. Any idea?
This is a contest math question that I don't remember the reference.
When does $f(x;\alpha)=\cos(\alpha x)-\sin^2x-1$ has unique zero?
Obviously, $f(0;\alpha)=0$ for all $\alpha\in{\mathbb R}$. I've no idea what theorem can be used here. Any idea?
HINT
$$\cos(\alpha x) - 1 -\sin^2(x) = 0 \implies \cos(\alpha x) = 1 + \sin^2(x) \geq 1$$
Hint Check this fact
$$ |\cos(x)| \leq 1 \,. $$