Cho biểu thức $P$ = \(\cos A + \cos B + \cos C\). Trong đó $A, B, C$ là các góc của tam giác $ABC$ bất kì. Chứng minh rằng $P$ đạt giá trị lớn nhất nhưng không đạt giá trị nhỏ nhất.
$P=$ \(\cos A + \cos B + \cos C = 2\cos \frac{{A + B}}{2}c{\rm{os}}\frac{{A - B}}{2} + 1 - 2{\sin ^2}\frac{C}{2}\)
    \(= 1 + 2\sin \frac{C}{2}c{\rm{os}}\frac{{A - B}}{2} + 1 - 2{\sin ^2}\frac{C}{2}\)
    \(=1 + 2\sin \frac{C}{2}\left( {c{\rm{os}}\frac{{A - B}}{2} - c{\rm{os}}\frac{{A + B}}{2}} \right) = 1 + 4\sin \frac{C}{2}\sin \frac{A}{2}\sin \frac{B}{2}\)
Từ đó: \(P = 1 - 2\left( {{{\sin }^2}\frac{C}{2} - \sin \frac{C}{2}c{\rm{os}}\frac{{A - B}}{2}} \right)\)
                \( = 1 - 2{\left( {\sin \frac{C}{2} - \frac{1}{2}c{\rm{os}}\frac{{A - B}}{2}} \right)^2} + \frac{1}{2}c{\rm{o}}{{\rm{s}}^2}\frac{{A - B}}{2}\)
                \( = \frac{3}{2} - 2{\left( {\sin \frac{C}{2} - \frac{1}{2}c{\rm{os}}\frac{{A - B}}{2}} \right)^2} - \frac{1}{2}{\sin ^2}\frac{{A - B}}{2} \le \frac{3}{2}\)
\(P = \frac{3}{2} \Leftrightarrow \left\{ \begin{array}{l}
\sin \frac{{A - B}}{2} = 0\\
\sin \frac{C}{2} - \frac{1}{2}c{\rm{os}}\frac{{A - B}}{2} = 0
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
A = B\\
\sin \frac{C}{2} = \frac{1}{2}
\end{array} \right. \Leftrightarrow \Delta ABC\)đều
Đáp số: $\max P$ = \(\frac{3}{2}\) xảy ra khi và chỉ khi \(\Delta ABC\)đều

Mặt khác \(P = 1 + 4\sin \frac{A}{2}.\sin \frac{B}{2}.\sin \frac{C}{2} > 1,\forall \Delta ABC\), do đó nếu $P$ đạt giá trị nhỏ nhất bằng $m$ thì ta cũng có \(P \ge m > 1,\forall \Delta ABC\)
Xét các tam giác $ABC$ với \(A = 2x,B = C = \frac{\pi }{2} - x\) thì \(P = 1 + 4\sin .{\sin ^2}\left( {\frac{\pi }{4} - \frac{x}{2}} \right);\mathop {\lim P}\limits_{x \to 0}  = \mathop {\lim }\limits_{x \to 0} \left( {1 + 4\sin x.{{\sin }^2}\left( {\frac{\pi }{4} - \frac{x}{2}} \right)} \right) = 1 + 0 = 1\)
Nhưng với P \( \ge m,\forall \Delta ABC \Rightarrow \mathop {\lim P}\limits_{x \to 0}  \ge m \Rightarrow 1 \ge m\) (vô lý) .
Vậy $P$ không có giá trị nhỏ nhất
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