Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
    
Not, I hope, an ongoing contest? –  Gerry Myerson Oct 10 '12 at 4:36
    
@GerryMyerson:No. I know the policy of math.SE and I've read this post: math.SE policy on question from ongoing contests on meta. If there is any unnecessary "confusion", I would vote to close this question. –  Jack Oct 10 '12 at 12:28
add comment

2 Answers

up vote 3 down vote accepted

HINT

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

share|improve this answer
    
Indulging in calculus for so long that I didn't notice how few facts I actually need to solve this "high school" problem. –  Jack Oct 10 '12 at 12:34
add comment

Hint Check this fact

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

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.