朱少卿 李雪芳

(廣東省佛山市高明區(qū)第一中學(xué))

1 題目呈現(xiàn)及解析

2 問題探究

由函數(shù)f(x)的圖像關(guān)于直線x＝a對(duì)稱可得f(a＋x)＝f(a-x),兩邊分別求導(dǎo)得f′(a＋x)＝-f′(a-x),即g(a＋x)＋g(a-x)＝0,所以函數(shù)g(x)的圖像關(guān)于點(diǎn)(a,0)對(duì)稱且g(a)＝0.因?yàn)間(2a＋x)＋g(-x)＝0,g(2b＋x)＝g(-x),所以g(2a＋x)＋g(2b＋x)＝0,將x換為x-2a＋2b,則g(2b＋x)＋g(4b-2a＋x)＝0,故g(4b-2a＋x)＝g(2a＋x),所以g(4b-4a＋x)＝g(x),函數(shù)g(x)為周期函數(shù)且周期T＝4|a-b|.

由函數(shù)g(x)的圖像關(guān)于直線x＝b對(duì)稱可得g(b＋x)＝g(b-x),構(gòu)造函數(shù)F(x)＝f(b＋x)＋f(b-x),顯然F′(x)＝f′(b＋x)-f′(b-x)＝g(b＋x)-g(b-x)＝0,所以F(x)≡c(c為常數(shù),令x＝0,則c＝2f(b)),所以f(b＋x)＋f(b-x)＝2f(b),函數(shù)f(x)的圖像關(guān)于點(diǎn)(b,f(b))對(duì)稱,函數(shù)f(x)為周期函數(shù)且周期T＝4|a-b|.

結(jié)論1 函數(shù)f(x)及其導(dǎo)函數(shù)f′(x)的定義域均為R,若f(x)和f′(x)的圖像分別關(guān)于直線x＝a和x＝b對(duì)稱,那么f(x)和f′(x)的圖像分別關(guān)于點(diǎn)(b,f(b))和點(diǎn)(a,0)對(duì)稱且f′(a)＝0,函數(shù)f(x)和f′(x)均為周期函數(shù)且周期T＝4|a-b|.

由探究1可知,函數(shù)g(x)的圖像關(guān)于點(diǎn)(a,0)對(duì)稱且g(a)＝0,因?yàn)間(2a＋x)＋g(-x)＝0,g(2b＋x)＋g(-x)＝0,所以g(2a＋x)＝g(2b＋x),所以g(2a-2b＋x)＝g(x),函數(shù)g(x)為周期函數(shù)且周期T＝2|a-b|.構(gòu)造函數(shù)F(x)＝f(b＋x)-f(bx),則F′(x)＝f′(b＋x)＋f′(b-x)＝g(b＋x)＋g(b-x),因?yàn)間(b＋x)＋g(b-x)＝0,即F′(x)＝0,所以F(x)≡c(c為常數(shù),令x＝0,則c＝0),所以f(b＋x)＝f(b-x),函數(shù)f(x)圖像關(guān)于直線x＝b對(duì)稱,函數(shù)f(x)為周期函數(shù)且周期T＝2|a-b|.

結(jié)論2 函數(shù)f(x)及其導(dǎo)函數(shù)f′(x)的定義域均為R,若f(x)和f′(x)的圖像分別關(guān)于直線x＝a和點(diǎn)(b,0)對(duì)稱,那么f(x)和f′(x)的圖像分別關(guān)于直線x＝b和點(diǎn)(a,0)對(duì)稱且f′(a)＝0,函數(shù)f(x)和f′(x)均為周期函數(shù)且周期T＝2|a-b|.

探究3 若將結(jié)論2中的“f′(x)的圖像關(guān)于點(diǎn)(b,0)對(duì)稱”改為“f′(x)的圖像關(guān)于點(diǎn)(b,t)(t≠0)對(duì)稱”,其他條件不變,函數(shù)f(x)的圖像是否關(guān)于直線x＝b對(duì)稱?

由函數(shù)f(x)的圖像關(guān)于點(diǎn)(a,0)對(duì)稱,可得

所以函數(shù)g(x)的圖像關(guān)于直線x＝a對(duì)稱,由探究2可知函數(shù)g(x)為周期函數(shù)且周期T＝2|a-b|.由探究1可知,函數(shù)f(x)圖像關(guān)于點(diǎn)(b,f(b))對(duì)稱,函數(shù)f(x)為周期函數(shù)且周期T＝2|a-b|.

結(jié)論3 函數(shù)f(x)及其導(dǎo)函數(shù)f′(x)的定義域均為R,若f(x)和f′(x)的圖像分別關(guān)于點(diǎn)(a,0)和直線x＝b對(duì)稱,那么f(x)和f′(x)的圖像分別關(guān)于點(diǎn)(b,f(b))和直線x＝a對(duì)稱,函數(shù)f(x)和f′(x)均為周期函數(shù)且周期T＝2|a-b|.

結(jié)論4 函數(shù)f(x)及其導(dǎo)函數(shù)f′(x)的定義域均為R,若f(x)和數(shù)f′(x)的圖像分別關(guān)于點(diǎn)(a,0)和點(diǎn)(b,0)對(duì)稱,那么f(x)和f′(x)的圖像分別關(guān)于直線x＝b和直線x＝a對(duì)稱,函數(shù)f(x)和f′(x)均為周期函數(shù)且周期T＝4|a-b|.

結(jié)論5 可導(dǎo)函數(shù)f(x)的圖像關(guān)于點(diǎn)(a,f(a))對(duì)稱的充要條件是其導(dǎo)函數(shù)f′(x)的圖像關(guān)于直線x＝a對(duì)稱,可導(dǎo)函數(shù)f(x)的圖像關(guān)于直線x＝a對(duì)稱的充要條件是其導(dǎo)函數(shù)f′(x)的圖像關(guān)于點(diǎn)(a,0)對(duì)稱.

結(jié)論6 若函數(shù)f(x)的圖像關(guān)于點(diǎn)(a,c)成中心對(duì)稱,關(guān)于直線x＝b成軸對(duì)稱,則f(x)是周期函數(shù)且周期T＝4|a-b|;若函數(shù)f(x)的圖像分別關(guān)于點(diǎn)(a,c)和點(diǎn)(b,c)成中心對(duì)稱,則函數(shù)f(x)是周期函數(shù)且周期T＝2|a-b|;若函數(shù)f(x)的圖像分別關(guān)于直線x＝a和直線x＝b成軸對(duì)稱,則函數(shù)f(x)是周期函數(shù)且周期T＝2|a-b|.

3 推廣應(yīng)用

例1 設(shè)函數(shù)f(x)＝x3-(a-1)x2＋ax-2,若f(x＋1)為奇函數(shù),則a的值是________.

因?yàn)閒(x＋1)為奇函數(shù),所以f(x)的圖像關(guān)于點(diǎn)(1,0)對(duì)稱,根據(jù)結(jié)論3或結(jié)論4可知其導(dǎo)函數(shù)f′(x)的圖像關(guān)于直線x＝1對(duì)稱.

例2 (2013年新課標(biāo)Ⅱ卷理10、文11)已知函數(shù)f(x)＝x3＋ax2＋bx＋c,下列結(jié)論中錯(cuò)誤的是( ).

A.?x0∈R,f(x0)＝0

B.函數(shù)y＝f(x)的圖像是中心對(duì)稱圖形

C.若x0是f(x)的極小值點(diǎn),則f(x)在區(qū)間(-∞,x0)上單調(diào)遞減

D.若x0是f(x)的極值點(diǎn),則f′(x0)＝0

對(duì)于選項(xiàng)C,由于f′(x)＝3x2＋2ax＋b的圖像是開口向上的二次函數(shù),若x0是f(x)的極小值點(diǎn),則f(x)在區(qū)間(-∞,x0)上不單調(diào)(先增后減),故選項(xiàng)C錯(cuò)誤.

對(duì)于選項(xiàng)D,若x0是f(x)的極值點(diǎn),根據(jù)導(dǎo)數(shù)的意義可知f′(x0)＝0,故選項(xiàng)D 正確.

綜上,選C.