Question

已知非负数x,y,z满足(x-1)/2=(2-y)/3=(z-3)/4，记w=3x+4y+5z,求w的最大值与最小值。

请帮我写出解答的过程。要完整的解答过程！谢谢

Answer 1

设(x-1)/2=(2-y)/3=(z-3)/4＝k
x-1=2k,x=2k+1
2-y=3k,y=2-3k
z-3=4k,z=4k+3
W=3(2k+1)+4(2-3k)+5(4k+3)
=6k+3+8-12k+20k+15
=6k+26
因为x>=0,y>=0,z>=0
2k+1>=0,k>=-0.5
2-3k>=0,k<=2/3
4k+3>=0,k>=-3/4
所以-0.5<=k<=2/3
Wmin=6*(-0.5)+26=23
Wmax=6*2/3+26=30

Answer 2

设(x-1)/2=(2-y)/3=(z-3)/4=k
x=2k+1,x>=0,k>=-1/2
y=2-3k,y>=0,k<=2/3
z=4k+3,z>=0,k>=-3/4

-1/2<=k<=2/3

w=3x+4y+5z
=3(2k+1)+4(2-3k)+5(4k+3)
=14k+26

k=2/3,w的最大值28/3+26=
k=-1/2,w的最小值19

Answer 3

设(x-1)/2=(2-y)/3=(z-3)/4＝k
x-1=2k,x=2k+1
2-y=3k,y=2-3k
z-3=4k,z=4k+3
W=3(2k+1)+4(2-3k)+5(4k+3)
=6k+3+8-12k+20k+15
=6k+26
因为x>=0,y>=0,z>=0
2k+1>=0,k>=-0.5
2-3k>=0,k<=2/3
4k+3>=0,k>=-3/4
所以-0.5<=k<=2/3
Wmin=6*(-0.5)+26=23
Wmax=6*2/3+26=30