Question

Помогите с заданиями, не сходится с ответом 1. Найти область значений функции: f(x) = 4cos²x - 4cosx + 1 2.Найти наибольшее значение функции: f(x) = 4sin2x + 4√3 cos2x 3.Указать множество значений функции: f(x) = 4cos3x·cos5x - 2cos2x + 11

Answer 1

1)  . Найти область значений функции:
f(x) = 4cos²x - 4cosx + 1, (2cox - 1)^2, с учётом IcosxI ≤ 1 составляем двойное неравенство и решив его, получаем:
min{4cos²x - 4cosx + 1} = 0, при x = - π/3 + 2πn и x π/3 + 2πn
max{4cos²x - 4cosx + 1} = 9, при x = - π + 2πn и x = π + 2πn
E(y) = [0 ; 9]
2)  Найти наибольшее значение функции:
 y = 4*sin(2*x)+4*(3^(1/2))*cos(2*x)
Находим первую производную функции:
y' = - 8√3*sin(2x) + 8*cos(2x)
Приравниваем ее к нулю:
 - 8√3*sin(2x) + 8*cos(2x) = 0
x1 = 1/12π
x2 = -1.31
Вычисляем значения функции 
f(1/12π) = 8
f(-1.31) = -3,46
Ответ: fmin = -3,46, fmax = 8
Используем достаточное условие экстремума функции одной переменной. Найдем вторую производную:
y'' = -16sin(2x) - 16√3cos(2x)
Вычисляем:
y''(1/12π) = -32 < 0 - значит точка x = 1/12π точка максимума функции.
y''(-1.31) = 8 > 0 - значит точка x = -1.31 точка минимума функции.
3)  Указать множество значений функции:
f(x) = 4cos3x·cos5x - 2cos2x + 11 с учётом IcosxI ≤ 1 составляем двойное неравенство и решив его, получаем:
E(y) = [9;13]

Answer 2

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Answer 3

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