Question

(理)給出下列四個命題:

①當x＞0且x≠1時,有lnx+≥2;②函數f(x)=lg(ax+1)的定義域是{x|x＞};

③函數f(x)=e^-xx²在x=2處取得極大值;④圓x²+y²-10x+4y-5=0上任意一點M關于直線ax-y-5a-2=0的對稱點M′也在該圓上.

所有正確命題的序號是__________.

Answer 1

解析：①x＞0,lnx∈R,∴lnx+≥2或lnx+≤-2,不正確;

②當a＞0時,定義域為{x|x＞},

當a＜0時,定義域為{x|x＜},不正確;

③f′(x)=-e^-xx²+e^-x·2x=-e^-x·x(x-2),

當x=2時,f′(x)=0;

當x∈(0,2)時,f′(x)＞0;當x∈(2,+∞)時,f′(x)＜0.∴x=2時,f(x)取極大值,正確;

④由x²+y²-10x+4y-5=0,得(x-5)²+(y+2)²=34,圓心為(5,-2),適合ax-y-5a-2=0.

∴ax-y-5a-2=0過圓心,故正確.

∴③④正確.

答案：③④