y=3sin^2(x)+6sin(x)cos(x)+5cos^2(x)
=3(sin^2(x)+cos^2(x))+3sin(2x)+1+cos(2x) ←２倍角の公式を適用
=4+3sin(2x)+cos(2x)
=4+(√10)sin(2x+a) ←合成公式を適用
ここで、cos(a)=3/√10, sin(a)=1/√10
π/10<a=tan^-1(1/3)<π/9
a≦2x+a<4π+aより
2x+a=π/2,5π/2,9π/2のとき y最大値=4+√10
x=π/4-(1/2)tan^-1(1/3),5π/4-(1/2)tan^-1(1/3),9π/4-(1/2)tan^-1(1/3)
2x+a=3π/2,7π/2のとき y最小値=4-√10
x=3π/4-(1/2)tan^-1(1/3),7π/4-(1/2)tan^-1(1/3)
