x>=0,y>= 0 x+2y=1/2 求函数log1/2(8xy+4y^2+1的最值
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x>=0,y>=0x+2y=1/2求函数log1/2(8xy+4y^2+1的最值x>=0,y>=0x+2y=1/2求函数log1/2(8xy+4y^2+1的最值x>=0,y>=0x+2y=1/2求函数
x>=0,y>= 0 x+2y=1/2 求函数log1/2(8xy+4y^2+1的最值
x>=0,y>= 0 x+2y=1/2 求函数log1/2(8xy+4y^2+1的最值
x>=0,y>= 0 x+2y=1/2 求函数log1/2(8xy+4y^2+1的最值
根据题意,得
x=1/2 -2y≥0,得y≤1/4
∵y≥0
∴y∈[0,1/4]
将x=1/2 -2y代入8xy+4y²+1,得
8xy+4y²+1
=8y(1/2 -2y)+4y²+1
=4y-12y²+1
=-12y²+4y+1
此函数图像的对称轴是y=1/6,开口向下
在y∈[0,1/4]范围内,
最大值在y=1/6处取得,是4/3
最小值在y=0处取得,是1
∴8xy+4y²+1∈[1,4/3]
∴log1/2(8xy+4y^2+1)∈[log2(3) -2,0]
即最大值是0
最小值是-2+log2(3)
其中,log2(3)指的是以2为底,3的对数
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