y=sinx+cos2x
=sinx+1-2sin²x
=-2sin²+sinx+1
=-2【sin²-2sinx×1/4+（1/4）²-（1/4）²】+1
=-2【(sinx-1/4)²-1/16】+1
=-2(sinx-1/4)²+1/8+1
=-2(sinx-1/4)²+9/8
≤9/8
所以最大值为9/8,此时sinx=1/4