0
$\begingroup$

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?

  • 0
    @GerryMyerson:No. I know the policy of math.SE and I've read this post: [math.SE policy on question from ongoing contests](http://meta.math.stackexchange.com/q/6206/9464) on meta. If there is any unnecessary "confusion", I would vote to close this question.2012-10-10

2 Answers 2

3

HINT

$\cos(\alpha x) - 1 -\sin^2(x) = 0 \implies \cos(\alpha x) = 1 + \sin^2(x) \geq 1$

  • 0
    Indulging in calculus for so long that I didn't notice how few facts I actually need to solve this "high school" problem.2012-10-10
1

Hint Check this fact

$ |\cos(x)| \leq 1 \,. $