Задача 71434 ...

63d507cefec4123cec4cd904

12.05.2023 20:08:44

Найдите наибольшее значение выражения 2/(√2/2 cosa + √2/2 sina + 3)

5f3ea7e3faf909182968ddd9

12.05.2023 20:14:57

★

[m]\frac{\sqrt{2}}{2}=sin\frac{π}{4}=cos\frac{π}{4}[/m]


[m]\frac{\sqrt{2}}{2}sin α +\frac{\sqrt{2}}{2}cos α =cos\frac{π}{4}sin α +sin\frac{π}{4}cos α=sin(\frac{π}{4}+ α) [/m]

[m]\frac{\sqrt{2}}{2}sin α +\frac{\sqrt{2}}{2}cos α+3=sin(\frac{π}{4}+ α) +3 [/m]


[m]- 1≤ sin(\frac{π}{4}+ α) ≤1 [/m]

[m]- 1+3≤ sin(\frac{π}{4}+ α) +3 ≤1+3 [/m] ⇒ [b]2[/b] [m]≤ sin(\frac{π}{4}+ α) +3 ≤ [/m][b]4[/b]


Наибольшее значение

[m]\frac{2}{2}=1[/m]