ta có :
\[P \ge \left( {\sin x + \cos x} \right)^3 + \frac{{16}}{{\left( {\sin x + \cos x} \right)^4 }} \ge \frac{{16 - 8\sqrt 2 }}{{\left( {\sin x + \cos x} \right)^4 }} +\frac{{8\sqrt2 }}{{\left( {\sin x + \cos x} \right)^4 }} + \left( {\sin x + \cos x} \right)^3 \]
mà \[\sin x + \cos x \le \sqrt 2 \]
\[\Rightarrow P \ge \frac{{16 - 8\sqrt 2 }}{{\left( {\sin x + \cos x} \right)^4 }} + \frac{8}{{\left( {\sin x + \cos x} \right)^3 }} + \left( {\sin x + \cos x} \right)^3 \]
Từ đây theo AM-GM
\[ \Rightarrow P \ge \frac{{16 - 8\sqrt 2 }}{{\left( {\sin x + \cos x} \right)^4 }} + 4\sqrt 2 \ge 4 + 2\sqrt 2 \]
Vậy Min\[P = 4 + 2\sqrt 2 \]
Dấu bằng xảy ra khi\[\sin x = \cos x = \frac{{\sqrt 2 }}{2}\]