【題目】【2017高考課標(biāo)2 文21】設(shè)函數(shù).

（1）討論的單調(diào)性；

（2）當(dāng)時，，求的取值范圍.

【答案】（Ⅰ）在 和單調(diào)遞減，在單調(diào)遞增（Ⅱ）

【解析】

試題分析：（1）先求函數(shù)導(dǎo)數(shù)，再求導(dǎo)函數(shù)零點(diǎn)，列表分析導(dǎo)函數(shù)符號確定單調(diào)區(qū)間（2）對分類討論，當(dāng)a≥1時，，滿足條件；當(dāng)時，取，當(dāng)0＜a＜1時，取，.

試題解析：（1）

令得

當(dāng)時，；當(dāng)時，；當(dāng)時，

所以在 和單調(diào)遞減，在單調(diào)遞增

(2)

當(dāng)a≥1時，設(shè)函數(shù)h(x)=（1-x）ex，h’(x)= -xex＜0（x＞0），因此h(x)在[0，+∞)單調(diào)遞減，而h(0)=1，

故h(x)≤1，所以

f(x)=（x+1）h(x)≤x+1≤ax+1

當(dāng)0＜a＜1時，設(shè)函數(shù)g（x）=ex-x-1，g’（x）=ex-1＞0（x＞0），所以g（x）在在[0，+∞)單調(diào)遞增，而g(0)=0，故ex≥x+1

當(dāng)0＜x＜1，，，取

則

當(dāng)時，取

綜上，a的取值范圍[1，+∞）

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