Question

(理)定義在R上的函數(shù)f(x)滿足,且函數(shù).給出下列結(jié)論:

①函數(shù)f(x)的最小正周期是;

②函數(shù)f(x)的圖象關(guān)于點(diǎn)(,0)對(duì)稱;

③函數(shù)f(x)的圖象關(guān)于直線x=對(duì)稱;

④函數(shù)f(x)的最大值為f().

其中正確結(jié)論的序號(hào)是_________.(寫出所有你認(rèn)為正確的結(jié)論的序號(hào))

Answer 1

(理)解析:由f(x+)+f(x)=0,得f(x+)=-f(x),

f(x+5)=-f(x+)=-［-f(x)］=f(x).

∴為周期函數(shù),最小正周期為5,①不正確;

∵f(x+)為奇函數(shù),關(guān)于原點(diǎn)對(duì)稱,

又f(x+)向右平移個(gè)單位得到f(x),

∴f(x)關(guān)于(,0)對(duì)稱,②正確;

∵f(x)=-f(x+)=-f(-x),

∴f(x+)=f(-x).

∴f(x)的圖象關(guān)于x=對(duì)稱,③正確;

f(x)的單調(diào)性無從考查,x=時(shí)不一定取最大值,④不正確.

∴②③正確.

答案:②③