Question

高数高手解答：设f(x,y)=(x^2)*y*(4-x-y),求在由x+y=4和x=0,y=0围成的闭区域内的最大值和最

Answer 1

f(x,y)=(4x^2-x^3-x^2y)*y

f'x=(8x-3x^2-2yx)*y

f'y=(4x^2-x^3)-2x^2y

f'x=0

8x-3x^2-2yx=0 2y=8-3x

f'y=0

4x^2-x^3-2x^2y=0 4-x-2y=0

x=2,y=1时,f'x=0,f'y=0,f(xy)=4

x+y=4交x=0于A(0,4) 交y=0于B(4,0)该闭区域内,

x+y≤4,　4-x-y≥0

x≠2,y≠1时

f'x=(8x-3x^2-2yx)*y=[-3(x-4-y)^2+64+y^2]y>0

f'y=(4x^2-x^3)-2x^2y=x^2(4-x-y)>0

x=2,y=1时,f(x,y)最大值=4

x=0,y=0时,f(x,y)最小值=0