數(shù)學(xué)學(xué)案（新人教B版）第三章37利用導(dǎo)數(shù)研究函數(shù)的零點(diǎn)

利用導(dǎo)數(shù)研究函數(shù)的零點(diǎn)考試要求函數(shù)零點(diǎn)問(wèn)題在高考中占有很重要的地位，主要涉及判斷函數(shù)零點(diǎn)的個(gè)數(shù)或范圍．高考?？疾槿魏瘮?shù)與復(fù)合函數(shù)的零點(diǎn)問(wèn)題，以及函數(shù)零點(diǎn)與其他知識(shí)的交匯問(wèn)題，一般作為解答題的壓軸題出現(xiàn)．題型一利用函數(shù)性質(zhì)研究函數(shù)的零點(diǎn)例1已知函數(shù)f(x)＝xsinx－1.(1)討論函數(shù)f(x)在區(qū)間eq\b\lc\[\rc\](\a\vs4\al\co1(－\f(π,2)，\f(π,2)))上的單調(diào)性；(2)證明：函數(shù)y＝f(x)在[0，π]上有兩個(gè)零點(diǎn)．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________思維升華利用函數(shù)性質(zhì)研究函數(shù)的零點(diǎn)，主要是根據(jù)函數(shù)單調(diào)性、奇偶性、最值或極值的符號(hào)確定函數(shù)零點(diǎn)的個(gè)數(shù)，此類(lèi)問(wèn)題在求解過(guò)程中可以通過(guò)數(shù)形結(jié)合的方法確定函數(shù)存在零點(diǎn)的條件．跟蹤訓(xùn)練1(2023·蕪湖模擬)已知函數(shù)f(x)＝ax＋(a－1)lnx＋eq\f(1,x)－2，a∈R.(1)討論f(x)的單調(diào)性；(2)若f(x)只有一個(gè)零點(diǎn)，求a的取值范圍．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________題型二數(shù)形結(jié)合法研究函數(shù)的零點(diǎn)例2(2023·鄭州質(zhì)檢)已知函數(shù)f(x)＝ex－ax＋2a，a∈R.(1)討論函數(shù)f(x)的單調(diào)性；(2)求函數(shù)f(x)的零點(diǎn)個(gè)數(shù)．________________________________________________________________________________________________________________________________________________________________________________________________________________________思維升華含參數(shù)的函數(shù)零點(diǎn)個(gè)數(shù)，可轉(zhuǎn)化為方程解的個(gè)數(shù)，若能分離參數(shù)，可將參數(shù)分離出來(lái)后，用x表示參數(shù)的函數(shù)，作出該函數(shù)的圖象，根據(jù)圖象特征求參數(shù)的范圍或判斷零點(diǎn)個(gè)數(shù)．跟蹤訓(xùn)練2(2023·長(zhǎng)沙模擬)已知函數(shù)f(x)＝alnx－2eq\r(x).(1)若a＝2，求曲線(xiàn)y＝f(x)在x＝1處的切線(xiàn)方程；(2)若函數(shù)f(x)在(0,16]上有兩個(gè)零點(diǎn)，求a的取值范圍．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________題型三構(gòu)造函數(shù)法研究函數(shù)的零點(diǎn)例3(12分)(2022·新高考全國(guó)Ⅰ)已知函數(shù)f(x)＝ex－ax和g(x)＝ax－lnx有相同的最小值．(1)求a；[切入點(diǎn)：求f(x)，g(x)的最小值](2)證明：存在直線(xiàn)y＝b，其與兩條曲線(xiàn)y＝f(x)和y＝g(x)共有三個(gè)不同的交點(diǎn)，并且從左到右的三個(gè)交點(diǎn)的橫坐標(biāo)成等差數(shù)列．[關(guān)鍵點(diǎn)：利用函數(shù)的性質(zhì)與圖象判斷ex－x＝b，x－lnx＝b的解的個(gè)數(shù)及解的關(guān)系]思維升華涉及函數(shù)的零點(diǎn)(方程的根)問(wèn)題，主要利用導(dǎo)數(shù)確定函數(shù)的單調(diào)區(qū)間和極值點(diǎn)，根據(jù)函數(shù)零點(diǎn)的個(gè)數(shù)尋找函數(shù)在給定區(qū)間內(nèi)的極值以及區(qū)間端點(diǎn)的函數(shù)值與0的關(guān)系，從而求得參數(shù)的取值范圍．跟蹤訓(xùn)練3(2021·全國(guó)甲卷)已知a>0且a≠1，函數(shù)f(x)＝eq\f(xa,ax)(x>0)．(1)當(dāng)a＝2時(shí)，求f(x)的單調(diào)區(qū)間；(2)若曲線(xiàn)y＝f(x)與直線(xiàn)y＝1有且僅有兩個(gè)交點(diǎn)，求a的取值范圍．______________________________________________________________________________________________________________________________________________________________________