導(dǎo)函數(shù)和原函數(shù)在對(duì)稱性上的聯(lián)系

支　軍

首先，我們來(lái)探究，原函數(shù)對(duì)稱時(shí)，導(dǎo)函數(shù)的對(duì)稱性如何？

若函數(shù)f(x)關(guān)于x=a對(duì)稱且可導(dǎo)，則f(x)=f(2a-x).

根據(jù)復(fù)合函數(shù)導(dǎo)數(shù)的性質(zhì)易得：

f ′(x)=-f ′(2a-x)，所以導(dǎo)函數(shù)f ′(x)關(guān)于點(diǎn)(a,0)對(duì)稱.

同理可得：若函數(shù)f(x)關(guān)于點(diǎn)(h,k)對(duì)稱且可導(dǎo)，則導(dǎo)函數(shù)f ′(x)關(guān)于直線x=h對(duì)稱.

因此，我們得到如下結(jié)論：

定理1 若函數(shù)f(x)關(guān)于x=a對(duì)稱且可導(dǎo)，則導(dǎo)函數(shù)f ′(x)關(guān)于點(diǎn)(a,0)對(duì)稱.若函數(shù)f(x)關(guān)于點(diǎn)(h,k)對(duì)稱且可導(dǎo)，則導(dǎo)函數(shù)f ′(x)關(guān)于直線x=h對(duì)稱.

推論 若函數(shù)f(x)為偶函數(shù)且可導(dǎo)，則導(dǎo)函數(shù)f ′(x)為奇函數(shù)；若函數(shù)f(x)為奇函數(shù)且可導(dǎo)，則導(dǎo)函數(shù)f ′(x)為偶函數(shù).

下面我們來(lái)探究導(dǎo)函數(shù)對(duì)稱時(shí)，原函數(shù)的對(duì)稱性如何.

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