x>0,y>0,且x分之一加y分之九等于1,求x加y的最小值

Question

x>0,y>0,且x分之一加y分之九等于1,求x加y的最小值

Answer 1

x+y最小为16,
由1/x + 1/y = 1 ==》y + 9x = xy;
==》y = 9x / (x - 1)
又因为：y > 0,
所以：x + y = 9x / (x - 1) +x ,且x > 1;
设 p = x - 1 ,则 p > 0;
x + y = [9(x-1)+9]/(x-1) + (x-1)+1 = (9p+9)/p + p+1;
==》x + y = 9/p + p + 10
设 m = 9/p + p 则 x+y = m +10 >10
由 m = 9/p + p 得：p*p -mp +9 = 0；
因为p有解,所以 m*m - 4*1*9 >= 0;
即：m*m - 36 >= 0
==》 m >= 6
即 x+y = m+10 >= 6+10 ==》x+y >=16
当x+y=16时,m=6,此时p=3,x=4,y=12
所以：当x=4,y=12时,x+y最小,为16