f(x)=(1+sinx)(1cosx)的最大最小值，定义域-30°≤x≤90°

题目是不是写错？

应该是 f(x)=(1+sinx)(1+cosx)吧

则f(x) = 1+sinx+cosx +sinxcosx

令t = sinx+cosx
则 t = √2sin(x+π/4)
√2sinπ/12≤t≤√2
由 (sinx+cosx)^2=(sinx)^2+2sinxcosx+(cosx)^2

所以 t^2 = 1+2sinxcosx
所以 sinxcosx = (t^2-1)/2
则
f(x) = 1+t +(t^2-1)/2 = 1/2(t^2+2t+1) = 1/2(t+1)^2

·当t = √2sinπ/12 时
f(x)取最小值 为 (sinπ/12)^2 + √2sinπ/12 +1/2

·当 t= √2 时
f(x)取最大值为 1/2(√2+1)^2 = 3/2 +√2