# the function $f(x)=x^2+2a\cos x-2bx$ with multiple critical points

Suppose that $f(x)=x^2+2a\cos x-2bx$, where $a,b$ constants.Prove that if $|a|>1$, the function has multiple critical points.

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Probably the question should be : Map regions of $(a,b)$ in $R^2$ where the equation has exactly $n$ roots. –  Maesumi Feb 14 '13 at 13:28

This does not seem correct. $f'(x)=2x-2a\sin x-2b=0$ could have just one root. graphing $y=x-1.1\sin x -2$ gives one root.