Question

已知非负数XYZ，X-1/2=2-Y/3=Z-3/4，记W=3X+4Y+5Z，求w的最大值和最小值

Answer 1

x-1/2=z-3/4 x=z-1/4
2-y/3=z-3/4 y=33/4-3z
w=3x+4y+5z
=3z-3/4+33-12z+5z
=-4z+32+1/4
∵x≥0 ∴z-1/4≥0 z≥1/4
y≥0 ∴33/4-3z≥0 -3z≥-33/4 z≤11/4
代入W等式，得：
W=-4×1/4+32+1/4=31+1/4
W=-4×11/4+32+1/4=21+1/4
W的最大值为31又1/4，最小值为21又1/4

Answer 2

令(x-1)/2=(2-y)/3=(z-3)/4=k
则
x=2k+1>0 k>-1/2
y=-3k+2>=0 k<=2/3
z=4k+3>=0 k>=-3/4
所以-1/2<=k<=2/3

W=3x+4y+5z
=3*(2k+1)+4*(-3k+2)+5*(4k+3)
=6k+3-12k+8+20k+15
=14k+26

当k=-1/2时w有最小值
W=14k+26=14*(-1/2)+26=-7+26=19

当k=2/3时w有最大值
W=14k+26=14*2/3+26=28/3+26=106/3