45. Найдите наибольшее значение выражения.
[tex] \sin( \frac{x}{2} ) \times { \cos( \frac{x}{2} ) }^{3} - \sin( \frac{x}{2} ) {}^{3 } \times \cos( \frac{x}{2} ) [/tex]
[tex] \sin( \frac{x}{2} ) \times { \cos( \frac{x}{2} ) }^{3} - \sin( \frac{x}{2} ) {}^{3 } \times \cos( \frac{x}{2} ) [/tex]
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[tex]\displaystyle\bf\\Sin\frac{x}{2}\cdot Cos^{3} \frac{x}{2} -Sin^{3} \frac{x}{2} \cdot Cos\frac{x}{2} =Sin\frac{x}{2} \cdot Cos\frac{x}{2} \cdot\underbrace{\Big(Cos^{2} \frac{x}{2} -Sin^{2} \frac{x}{2} \Big)}_{Cosx}=\\\\\\=\frac{1}{2} \cdot \underbrace{2Sin\frac{x}{2} Cos\frac{x}{2} }_{Sinx}\cdot Cosx=\frac{1}{2} Sinx Cosx =\frac{1}{2} \cdot \frac{1}{2} \cdot \underbrace{2Sinx Cosx}_{Sin2x}=\frac{1}{4} Sin2x\\\\\\-1\leq \frac{1}{4} Sin2x\leq 1\\\\-\frac{1}{4} \leq Sin2x\leq \frac{1}{4}[/tex]
[tex]\displaystyle\bf\\Otvet : \frac{1}{4}[/tex]