Question

已知常数a,b和正变量x,y满足a+b=10,a/x+b/y=1,x+y的最小值为18,求a,b的值

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Answer 1

x+y=(a/x+b/y)*(x+y)=a+b+ay/x+bx/y=10+ay/x+bx/y
(a/x+b/y)*(x+y)≥1*18=18
10+ay/x+bx/y≥18,ay/x+bx/y≥8
ay/x+bx/y≥2√(ay/x*bx/y)=2√ab
2√ab=8,ab=16
a+b=10,
a=2,b=8
a=8,b=2