Question

(理)給出下列四個(gè)命題:

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

③函數(shù)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(diǎn)M關(guān)于直線ax-y-5a-2=0的對(duì)稱點(diǎn)M′也在該圓上.

所有正確命題的序號(hào)是__________.

Answer 1

解析：①x＞0,lnx∈R,∴l(xiāng)nx+≥2或lnx+≤-2,不正確;

②當(dāng)a＞0時(shí),定義域?yàn)閧x|x＞},

當(dāng)a＜0時(shí),定義域?yàn)閧x|x＜},不正確;

③f′(x)=-e^-xx²+e^-x·2x=-e^-x·x(x-2),

當(dāng)x=2時(shí),f′(x)=0;

當(dāng)x∈(0,2)時(shí),f′(x)＞0;當(dāng)x∈(2,+∞)時(shí),f′(x)＜0.∴x=2時(shí),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