Question

函数y=－cos2x＋AcosX－A的最大值为1,求实数A

Answer 1

y = -(1/2)cos2x + Acosx -(1/2)A = -(1/2)(2cosx - 1) + Acosx - 1/2A = -cosx + Acosx + 1/2 - (1/2)A = -[cosx - Acosx + (-A/2) - (-A/2)] + (1/2)(1 - A) = -(cosx - A/2) + (1/4)A + (1/2)(1 - A) 因此,当cosx = -A/2时,y有极大值,即 (1/4)A + (1/2)(1 - A) = 1 (A - 1) = 3 A = 1±√3【注意到,|cosx|≤1,所以|A/2|≤1,故舍弃A = 1 + √3】 由此得到 A = 1 - √3