Question

函数最大值

实数x,y满足4x^2+3y^2=3x,则y^2+3x的最大值为

Answer 1

4x^2+3y^2=3x
∴4x^2+4y^2=3x+y^2
∴4（x^2+y^2）=3x+y^2
∵y^2=x-4/3x^2
∴4(-1/3x^2+x)=3x+y^2
∴只要-1/3x^2+x有最大值，则3x+y^2有最大值
-1/3x^2+x的最大值是当x取自变量最大值时
∵0≤x≤3/4
∴x=3/4
∴3x+y^2最大值为9/4

Answer 2

4x^2+3y^2=3x
y^2=x-(4/3)x^2大于等于0
x大于等于0小于等于3/4

y^2+3x=x-(4/3)x^2+3x=-(4/3)x^2+4x
x=3/4 y^2+3x的最大值为9/4

Answer 3

由4x^2+3y^2=3x 得y^2=x-4/3x^2
代入要求的式子=-4/3x^2+4x =-4/3(x-3/2)^2+3
但注意x有范围 x-4/3x^2≥0 0≤x≤3/4
所以x=3/4 y=0时 式子有最大值=9/4

Answer 4

y^2=-4/3x^2+x>=0,得0<=x<=3/4,y^2+3x=-4/3x^2+4x=-4/3(x-3/2)^2+3.当x=3/4时,y^2+3x取最大值9/4.