3sin＾2a+2sin＾2b=2sina求sin^2a+sin^2b的最大值

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3sin＾2a+2sin＾2b=2sina求sin^2a+sin^2b的最大值
3sin＾2a+2sin＾2b=2sina
求sin^2a+sin^2b的最大值

3sin＾2a+2sin＾2b=2sina求sin^2a+sin^2b的最大值
3sin＾2a+2sin＾2b=2sina
2*(sin^2a+sin^2b)=2sina-sin^2a
sin^2a+sin^2b=sina-1/2*sin^a
令sina=x(-1<=x<=1)
则sin^2a+sin^2b=sina-1/2*sin^a=-1/2x^2+x
二次函数的性质,
开口向下,
当x=1时,取最大值.
=-1/2x^2+x=-1/2+1=1/2
所以sin^2a+sin^2b的最大值是1/2

4
sin^2a+sin^2b
=(sin^a)^2+(sin^b)^2
=(sina+sinb)(sina-sinb)
因为sin的值域为（-1，1）
所以原式最大值=2*2=4

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因为sin^2b>=0,sin^2a>=0;
所以：2sin^2b+3sin^2a>=0.
即：2sina>=0,得到：sina>=0.

∵2sin^2b+3sin^2a=2sina，
∴sin^2b=sina-(3/2)sin^2a>=0
进一步：
Sina(1-3/2sina)>=0
1-3/2sina>=0,得到：sina<=2/3.
即sina的取值范围为：[0,2/3].
则：
m=sin^2a+sin^2b
=sin^2a+sina-(3/2)sin^2a
=-(1/2)sin^2a+sina
=-(1/2)(sin^2a-2sina+1)+1/2
=-(1/2)(sina-1)^2+1/2.
因为0<=sina<=2/3.所以：
当sina=2/3，m有最大值m=4/9
当sina=0，m有最小值m=0。
所以m的取值范围为：[0,4/9].

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