已知函数y=(sinx+2)(cosx+2)
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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
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
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