已知函数y=(sinx+2)(cosx+2)

y=sinxcosx+2(sinx+cosx)+4
令a=sinx+cosx=√2sin(x+π/4)
-π/2<=x<=π/2
-π/4<=x+π/4<=3π/4
所以sin(-π/4)<=sin(x+π/4)<=sin(π/2)
-√2/2<=sin(x+π/4)<=1
所以-1<=a<=√2

a=sinx+cosx
a²=sin²x+2sinxcosx+cos²a=1+2sinxcosx
sinxcosx=(a²-1)/2
所以y=(a²-1)/2+2a+4
=(1/2)(a+2)²+3/2
-1<=a<=√2
所以在对称轴a=-2右边，开口向上，所以是增函数
所以a=-1,y最小=2
a=√2,y最大=2√2+9/2

y=(sinx+2)(cosx+2)=sinxcosx+2(sinx+cosx)+4
y'=(cosx+sinx)(cosx-sinx)+2(cosx-sinx)=(cosx-sinx)(cosx+sinx+2)
x∈[-π/2,π/2]
x∈[-π/2,π/4),y'>0
x∈(π/4,π/2),y'<0

最大值 f(π/4)=2√2+9/2
f(-π/2)=7/2;f(π/2)=6;最小值f(-π/2)=7/2

化简 原式=sinxcosx + 2（sinx + cosx） + 4
=1/2 sin2x + 2√2sin（x+π/4） + 4
=-1/2cos[2(x+π/4)] + 2√2sin（x+π/4）+ 4
=sin^2（x+π/4）+ 2√2sin（x+π/4）+ 7/2
=[sin（x+π/4)+ √2 ]^2 + 3/2
由于 x∈[-π/2,π/2]