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I want to minimize a real valued function $f(x) = \frac{2}{3} \cos(x)+1+ \frac{1}{9}\cos^2(x),$ but it turns out that the minimum occurs at $x = \pi - \arccos 3$.

I am not sure what to conclude from this fact. Does the minimum now occur at $x = \pi$, or how should I interpret the fact that $\arccos$ is only defined in the interval $[-1,1]$?

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    @Gigili: What can we interpret about this problem when we see the graph of the function shows infinitely maximums?2012-06-03

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If we take $f'(x)=-\frac 2 3 \sin x - \frac 2 9 \cos x \sin x$ and set it equal to zero, we find that all critical points occur when $\sin x =0$ or $\cos x = -3$. The latter is impossible (over the reals), so all critical points occur when $\sin x =0$. This implies that $x=n \pi$, where $n\in \mathbb Z$. Examining the original equation, it is clear that the minimums occur when $n$ is odd or, equivalently, for $x=2k\pi+\pi$ for $k\in\mathbb Z$.

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Assuming the cosines are on the numerator:$f'(x)=-\frac{\sin x+6\sin x\cos x}{27}=0\Longleftrightarrow x=k\pi\,,\,k\in\mathbb{Z}\,,\,\,or\,\,\cos x=-\frac{1}{6}$I can't see how you got $\,\arccos 3\,$ there...

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    Indeed so. Thanks2012-06-03