Question

已知a>0,X∈[0,2π）,y=1/2cos2x-asinx+b+1/2的最大值为0 最小值为—4 求实数a,b的值

Answer 1

y = (1/2)cos2x - asinx + b + 1/2

= (1/2)(1 - 2sin²x) - asinx + b + 1/2

= -sin²x - asinx + b + 1

= -(sinx + a/2)² + a²/4 + b + 1

当 0 ≤ a ≤ 2 时,

a²/4 + b + 1 = 0

-1 - a + b + 1 = -4

解得: a = 2 , b = -2

当 -2 ≤ a ≤ 0 时,

a²/4 + b + 1 = 0

-1 + a + b + 1 = -4

解得: a = -2 , b = -2

当 a < -2 时,

-1 - a + b + 1 = 0

-1 + a + b + 1 = -4

无解

当 a > 2 时,

-1 + a + b + 1 = 0

-1 - a + b + 1 = -4

无解

综上: a = 2 , b = -2 或 a = -2 , b = -2