Let the minimum value of the function y = 2 (cosx) 2-2acosx - (2a + 1) about X be 0.5, and find the maximum value

Let the minimum value of the function y = 2 (cosx) 2-2acosx - (2a + 1) about X be 0.5, and find the maximum value

Cosx belongs to [- 1,1]; the opening of parabola is upward, and the axis of symmetry cosx = A / 2. When a / 21, the minimum value is Ymin = 2-2a-2a-1 = 1-4a = 0.5, a = 1 / 8, which does not satisfy a / 2 > 1. Only a / 2 belongs to [- 1,1]. When cosx = A / 2, the minimum value is Ymin = a ^ 2 / 2-A ^ 2-2a-1 = 0.5A ^ 2 + 2A + 3 = 0

Let the minimum value of the function y = 2 (cosx) ^ 2-2acosx - (2a + 1) of X be g (a). The number of zeros of the minimum function f (a) is discussed and proved

Y = 2 (cosx) ^ 2-2acosx - (2a + 1) = 2 (COS & # 178; x-acosx + A & # 178 / / 4) - (A & # 178 / / 2 + 2A + 1) = 2 (cosx-a / 2) &# 178; - (A & # 178 / / 2 + 2A + 1) this is a quadratic function of cosx, - 1 ≤ cosx ≤ 1. When a / 22, cosx = 1, y gets the minimum value of 1-4a {1, (A2) when A2, from the solution of 1-4a = 0, we get that