Question

若x乘以y是正实数,则(x+1/2y)^2+(y+1/2x)^2的最小值是?

Answer 1

设u=x+y,v=xy,

w=[x+1/(2y)]^2+[y+1/(2x)]^2

=x^2+x/y+1/(4y^2)+y^2+y/x+1/(4x^2)

=(x^2+y^2)[1+1/(xy)+1/(2xy)^2]

=(u^2-2v)[1+1/v+1/(4v^2)]

=(u^2-2v)[1+1/(2v)]^2,

0=4,

当u^2=2,v=1/2时取等号,

∴w的最小值是4,为所求.