已知x,y∈R+,2x+5y=10,求xy的最大值及相应x,y的值

题目

已知x,y∈R+,2x+5y=10,求xy的最大值及相应x,y的值

答案

2x+5y=10,y = (10 - 2x)/5
xy = x(10 - 2x)/5 = -2x²/5+ 2x = (-2/5)(x² -5x)
= (-2/5)[(x- 5/2)² - 25/4]
= (-2/5)(x - 5/2)² + 5/2
x = 5/2 (此时y = 1)时,xy取最大值5/2