<s>[INST] Minimum Value of $x_1+x_2+x_3$ For an Acute Triangle $\Delta ABC$
$$\begin{align}x_n=2^{n-3}\left(\cos^nA+\cos^nB+\cos^nC\right)+\cos A\,\cos B\,\cos C\end{align}$$ Then find the least value of $$x_1+x_2+x_3$$
My Approach: I have found $x_1$, $x_2$ and $x_3$ 
$$\begin{align}x_1=\frac{1}{4}\left(\cos A+\cos B+\cos C\right)+\cos A\,\cos B\,\cos C\\
 =\frac{1}{4}\left(1+4\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}\right)+\cos A\,\cos B\,\cos C \tag{1}\end{align}$$
$$\begin{align}x_2=\frac{1}{4}\left(3+\cos 2A+\cos 2B+\cos 2C\right)+\cos A\,\cos B\,\cos C=\frac{1}{2} \tag{2}\end{align}$$ 
$$x_3=\frac{1}{4}\left(3\cos A+3\cos B+3\cos C+\cos 3A+\cos 3B+\cos 3C\right)+\cos A\,\cos B\,\cos C$$ $$\implies x_3=\frac{1}{2}+x_1+\frac{1}{4}\sum \cos 3A+2\prod \sin\frac{A}{2}\\
 $$
$$\implies x_3=\frac{1}{2}+x_1-\prod \sin\frac{3A}{2}+2\prod \sin\frac{A}{2}\\
 \tag{3}$$
$$\text{So}\;\;\;\;\;\;\;\begin{align}x_1+x_2+x_3=\frac{3}{2}+4\prod \sin\frac{A}{2}-\prod \sin\frac{3A}{2}+2\prod \cos A \end{align}$$
I cannot proceed any further.
 [/INST] Use AM-GM inequality,we have
$$\cos^3{x}+\dfrac{\cos{x}}{4}\ge 2\sqrt{\cos^3{x}\cdot\dfrac{\cos{x}}{4}}=\cos^2{x}$$
then we have 
$$x_{1}+x_{3}\ge\cos^2{A}+\cos^2{B}+\cos^2{C}+2\cos{A}\cos{B}\cos{C}=2x_{2}$$
so
$$x_{1}+x_{2}+x_{3}\ge 3x_{2}=\dfrac{3}{2}$$
because we have use this follow well know
$$\cos^2{A}+\cos^2{B}+\cos^2{C}+2\cos{A}\cos{B}\cos{C}=1$$
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