Y=26100-XZ=(1400+X*3)（1.4+(X/100+Y/20)/100）问：当X取值多少时,Z有最大值我的提问没错，有知道答案的么？

Question

Y=26100-X
Z=(1400+X*3)（1.4+(X/100+Y/20)/100）
问：
当X取值多少时,Z有最大值
我的提问没错，有知道答案的么？

Answer 1

z = (1400 + x³)[1.4 + (x/100 + y/20)/100]
= (1400 + x³)[1.4 + x/10000 + y/2000]
= (1400 + x³)[1.4 + x/10000 + (26100 - x)/200]
= (1400 + x³)[14.45 - x/2500]
= 20230 - 0.56x + 14.45x³ - x⁴/2500
dz/dx = -0.56 + 43.35x² - x³/625
d²z/dx² = 86.7x - 3x²/625
令 dz/dx = 0,
解得：x₁= 27093.75,x₂= 0.11366,x₃= -0.11366
当 x₁= 27093.75,d²z/dx² = -1174514.063 < 0
当 x₂= 0.11366,d²z/dx² = 9.8543 〉 0
当 x₃= -0.11366,d²z/dx² = -0.9854 < 0
当 x = 27093.75,
z = [20230 - 0.56x + 14.45x³ - x⁴/2500]|(x=27093.75) = 7.1848×10¹³
当 x = -0.11366
z = [20230 - 0.56x + 14.45x³ - x⁴/2500]|(x=-0.11366) = 20230.042
所以,当 x = 27093.75,z(max) = 7.1848×10¹³