∵x,y屬于正實(shí)數(shù),x＋y＝1
∴0＜x＜1,0＜y＜1
令w＝xy＝x(1－x),則有
w＝－x²＋x＝－(x－1/2)²＋1/4
∴w在[0,1/2]上單調(diào)遞增,在[1/2,1]上單調(diào)遞減
∴當(dāng)x＝1/2時(shí),w＝1/4為最大值
假如x可以取到0或1,則w＝0為最小值
∴w∈(0,1/4]
z＝(x＋1/x)(y＋1/y)
＝xy＋1/(xy)＋x/y＋y/x
＝xy＋1/(xy)＋(x²＋y²)/(xy)
＝xy＋1/(xy)＋[(x＋y)²－2xy]/(xy)
＝xy＋1/(xy)＋(1－2xy)/(xy)
＝xy＋2/(xy)－2
＝w＋2/w－2
∵w＞0
∴w＋2/w≥2(w·2/w)^1/2＝2√2
且當(dāng)且僅當(dāng)w＝2/w,即w＝√2時(shí)等號(hào)成立
而√2不在w的取值范圍(0,1/4]內(nèi)
∴當(dāng)w＝1/4時(shí),z＝w＋2/w－2取得最小值,為25/4