Bài 112258

Question

a) Chứng minh rằng :

$\left ( \frac{\sin x }{x} \right )^3 > \cos x , \forall x \in \left ( 0;\frac{\pi}{2} \right ) .$
b) Tìm giá trị lớn nhất của biểu thức :

$P = \frac{1}{\sin ^2x} + \frac{1}{\sin ^2 y}+\frac{1}{\sin^2z}- \left ( \frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2} \right ) ; x,y,z \in \left ( 0;\frac{\pi}{2} \right )$

score 0 · Answer 1

a) Ta có :

$\cos x < 1 , \forall x \in \left ( 0;\frac{\pi}{2} \right ] \Rightarrow \int\limits_{0}^{t}\cos xdx < \int\limits_{0}^{t}dt, \forall t \in \left ( 0;\frac{\pi}{2} \right ]$

$\Rightarrow \sin t < t, \forall t \in \left ( 0;\frac{\pi}{2} \right ] \Rightarrow \int\limits_{0}^{x} \sin tdt < \int\limits_{0}^{x}tdt, \forall t \in \left ( 0;\frac{\pi}{2} \right ]$

$\Rightarrow 1 - \cos x < \frac{x^2}{2} , \forall x \in \left ( 0;\frac{\pi}{2} \right ]$

$\Rightarrow \int\limits_{0}^{t} ( 1 - \cos x)dx < \int\limits_{0}^{t} \frac{x^2}{2}dx , \forall t \in \left ( 0;\frac{\pi}{2} \right ]$

$\Rightarrow t - \sin t < \frac{t^3}{6}, \forall t \in \left ( 0;\frac{\pi}{2} \right ] \Rightarrow \sin t > t -\frac{t^3}{6}, \forall t \in \left ( 0;\frac{\pi}{2} \right ]$ (1)

$\Rightarrow \int\limits_{0}^{x}\sin tdt > \int\limits_{0}^{x} \left ( t-\frac{t^3}{6} \right )dt, \forall x \in \left ( 0;\frac{\pi}{2} \right ]$

$\Rightarrow 1 - \cos x > \frac{x^2}{2} - \frac{x^4}{24} , \forall x \in \left ( 0;\frac{\pi}{2} \right ]$

$\Rightarrow \cos x < 1 -\frac{x^2}{2} +\frac{x^4}{24}, \forall x \in \left ( 0;\frac{\pi}{2} \right ]$ (2)
Từ (1) và (2)

$\Rightarrow \left ( \frac{\sin x}{x} \right ) ^3 > \left ( 1 - \frac{x^2}{6} \right )^3 \geq 1 - \frac{x^2}{2}+\frac{x^4}{12}-\frac{x^6}{216}$

$> 1 - \frac{x^2}{2} +\frac{x^4}{24} > \cos x , \forall x \in \left ( 0;\frac{\pi}{2} \right ]$
Vậy :

$\left ( \frac{\sin x}{x} \right )^3 > \cos x, \forall x \in \left ( 0;\frac{\pi}{2} \right ]$ (3)
b) Từ (3)

$\Rightarrow \frac{\cos x }{\sin ^3 x} < \frac{1}{x^3} , \forall x \in \left ( 0;\frac{\pi}{2} \right ]$

$\Rightarrow \int\limits_{x}^{\frac{\pi}{2} } \frac{\cos t}{\sin ^3t}dt \leq \int\limits_{x}^{\frac{\pi}{2} } \frac{dt}{t^3}, \forall x \in \left ( 0;\frac{\pi}{2} \right ]$