# How to find minimum?

Is it possible to find exactly (not numerically) the minimum of the function $$\sqrt{15-12\cos(x)}+\sqrt{4-2\sqrt{3}\sin(x)}+\sqrt{7-4\sqrt{3}\sin(x)}+\sqrt{10-4\sqrt{3}\sin(x)-6\cos(x)}$$ on the interval $[0,2\pi)$ ? Maple 18 and Mathematica 10 fail with it.

• The derivative of the function cancels at $x=\pi/3$ and the minimum is exactly $6$. The big question is how to prove it. Oct 26, 2014 at 10:23
• @Claude Leibovici: Thank you for your interest to the question. Oct 26, 2014 at 10:26

• Numerically, the minimum is attained at approximately $1.5599$ (which is suspiciously close to, but clearly diffrent from $\frac\pi 2$) with a value of about $6.6248$. Oct 26, 2014 at 10:24
• Anwer is 6 ($x=pi/3$) Oct 26, 2014 at 10:27