Question

函数f(x)=cos^2x+sinxcosx的最大值 求解 详细过程 谢谢

Answer 1

cos^2x=（cos2x+1）/2
sinxcosx=sin2x/2
所以f(x)=（cos2x+sin2x+1）/2=(2^(1/2)sin(x+45°)+1)/2
所以最大值为(2^(1/2)+1)/2

Answer 2

令f′（x）=-2sinxcosx+cosxcosx-sinxsinx=cos(2x)-sin(2x)=0,则x=π/8或x=5π/8.
f（π/8）=cos(π/8)cos(π/8)+sin(π/8)cos(π/8)=[cos(π/4)+1]/2+sin(π/4)/2=(√2+2)/4+√2/4=√2/2+1，f（5π/8)=[cos(5π/4)+1]/2+sin(5π/4)/2=(2-√2)/4-√2/4=(1-√2)/2。√2/2+1＞(1-√2)/2，所以函数最大值为√2/2+1。

Answer 3

(cos2x+1)/2+(1/2)sin2x=（√2/2）sin(2x+排/4)+1/2
所以f(x)max=(√2+2)/2

Answer 4

(√2+1)/2