雙圖法讓解題更直觀自然

PAGE/r/nPAGE/r/n6/r/n“/r/n雙圖法/r/n”/r/n/r/n讓解題/r/n更直觀/r/n自然/r/n524500/r/n廣東省吳川市第一中學(xué)/r/n/r/n柯厚寶/r/n/r/n導(dǎo)數(shù)是研究函數(shù)的單調(diào)性、極值、最值問題的有力工具，/r/n也是高考的熱/r/n點(diǎn)/r/n./r/n解決該類問題的常用方法是先求函數(shù)/r/n的導(dǎo)數(shù)/r/n，通過解不等式/r/n與/r/n確定/r/n的單調(diào)區(qū)間，進(jìn)而求/r/n極值、最值/r/n./r/n而在實(shí)際操作中，常會在解不等式時(shí)遇到麻煩，/r/n特別是所求的不等式中涉及參數(shù)/r/n討論時(shí)/r/n，則更難處理/r/n./r/n本文/r/n運(yùn)用一種/r/n直觀的方法/r/n——“/r/n雙圖法/r/n”/r/n，/r/n解決此類問題/r/n./r/n/r/n所謂的/r/n“/r/n雙圖法/r/n”/r/n，就是/r/n運(yùn)用導(dǎo)數(shù)研究函數(shù)/r/n的/r/n單調(diào)性/r/n時(shí)，/r/n由于/r/n其單調(diào)性與導(dǎo)數(shù)/r/n的/r/n正、負(fù)/r/n密切相關(guān)/r/n，/r/n于是我們可以直接畫出/r/n(/r/n或/r/n一個(gè)與/r/n的正、負(fù)相關(guān)的函數(shù)/r/n)/r/n的/r/n圖象，根據(jù)其正、負(fù)再/r/n在下方/r/n描繪出/r/n的大致圖象，然后/r/n直觀地/r/n求/r/n得/r/n的單調(diào)區(qū)間、極值與最值/r/n的一種方法/r/n./r/n1/r/n．/r/n直觀/r/n探討/r/n函數(shù)的單調(diào)性/r/n/r/n求函數(shù)的單調(diào)區(qū)間時(shí)，/r/n我們/r/n只需同時(shí)描繪出/r/n(/r/n或與/r/n正、負(fù)相關(guān)的函數(shù)/r/n)/r/n與/r/n的圖象，即可直觀地得到/r/n的單調(diào)區(qū)間/r/n./r/n例/r/n1/r/n．/r/n求下列函數(shù)的單調(diào)區(qū)間：/r/n(/r/n1/r/n)/r/n；/r/n/r/n/r/n(/r/n2/r/n)/r/n，/r/n(/r/n3/r/n)/r/n；/r/n/r/n(/r/n4/r/n)/r/n./r/n圖1/r/n1/r/n3/r/n解析：/r/n(/r/n1/r/n)/r/n可得/r/n圖1/r/n1/r/n3/r/n與/r/n的圖象如圖所示/r/n1/r/n，/r/n當(dāng)/r/n或/r/n時(shí)，/r/n，/r/n單調(diào)遞增；當(dāng)/r/n時(shí)，/r/n，/r/n單調(diào)遞減；/r/n故/r/n的遞增區(qū)間為/r/n；遞減區(qū)間為/r/n./r/n(/r/n2/r/n)/r/n∵/r/n，/r/n，/r/n，/r/n令/r/n，/r/n，/r/n圖2/r/n=1\*GB3/r/n①/r/n當(dāng)/r/n時(shí)，/r/n，/r/n，/r/n在/r/n上是增函數(shù)，在/r/n上也是增函數(shù)，/r/n圖2/r/n=2\*GB3/r/n②/r/n當(dāng)/r/n時(shí)，/r/n令/r/n得/r/n，/r/n，/r/n與/r/n的圖象如圖所示/r/n2/r/n，/r/n當(dāng)/r/n或/r/n時(shí)，/r/n，/r/n，/r/n單調(diào)遞增，/r/n當(dāng)/r/n或/r/n時(shí)，/r/n，/r/n，/r/n單調(diào)遞減，/r/n綜上，當(dāng)/r/n時(shí)，/r/n的遞/r/n增/r/n區(qū)為/r/n，/r/n遞減區(qū)為/r/n；當(dāng)/r/n時(shí)，/r/n的遞增區(qū)為/r/n，遞減區(qū)為/r/n./r/n(3)/r/n，/r/n，/r/n由/r/n得/r/n，/r/n令/r/n，/r/n由/r/n得/r/n或/r/n，/r/n圖/r/n3/r/n=1\*GB3/r/n①/r/n當(dāng)/r/n，即/r/n時(shí)，/r/n，/r/n，/r/n的單調(diào)遞增區(qū)間是/r/n./r/n圖/r/n3/r/n=2\*GB3/r/n②/r/n當(dāng)/r/n，即/r/n時(shí)，/r/n與/r/n的圖象如圖/r/n3/r/n所示，/r/n當(dāng)/r/n或/r/n時(shí)，/r/n，/r/n，/r/n單調(diào)遞增，當(dāng)/r/n時(shí)，/r/n，/r/n，/r/n單調(diào)遞減，/r/n圖/r/n4/r/n=3\*GB3/r/n③/r/n當(dāng)/r/n，即/r/n時(shí)，/r/n與/r/n的圖象如圖/r/n4/r/n所示，/r/n圖/r/n4/r/n當(dāng)/r/n或/r/n時(shí)，/r/n，/r/n，/r/n單調(diào)遞增，/r/n當(dāng)/r/n時(shí)，/r/n，/r/n，/r/n單調(diào)遞減，/r/n綜上，當(dāng)/r/n時(shí)，/r/n的單調(diào)遞增區(qū)間是/r/n；當(dāng)/r/n時(shí)，/r/n的單調(diào)遞增區(qū)間是/r/n和/r/n，/r/n遞減區(qū)間是/r/n；/r/n當(dāng)/r/n時(shí)，/r/n的單調(diào)遞增區(qū)間/r/n是/r/n和/r/n，/r/n遞減區(qū)間是/r/n./r/n(4)/r/n，/r/n，/r/n令/r/n，/r/n，/r/n=1\*GB3/r/n①/r/n當(dāng)/r/n時(shí)，/r/n，/r/n，/r/n在/r/n上單調(diào)遞增，/r/n=2\*GB3/r/n②/r/n當(dāng)/r/n時(shí)，/r/n令/r/n，/r/n由/r/n得/r/n，由/r/n得/r/n，由/r/n得/r/n，/r/n(/r/n=1\*roman/r/ni/r/n)/r/n若/r/n時(shí)，/r/n，/r/n，僅當(dāng)/r/n時(shí)取等號，而/r/n，有/r/n，/r/n∴/r/n在/r/n上單調(diào)遞增，/r/n(/r/n=2\*roman/r/nii/r/n)/r/n若/r/n時(shí)，/r/n，/r/n，/r/n在/r/n上單調(diào)遞增，/r/n圖5/r/n(/r/n=3\*roman/r/niii/r/n)/r/n若/r/n時(shí)，由/r/n得/r/n，/r/n，/r/n圖5/r/n與/r/n的圖象如圖/r/n5/r/n所示，/r/n當(dāng)/r/n時(shí)，/r/n，/r/n，/r/n在/r/n上單調(diào)遞增，/r/n=3\*GB3/r/n③/r/n當(dāng)/r/n時(shí)，令/r/n，由/r/n得/r/n，/r/n由/r/n得/r/n，由/r/n得/r/n，/r/n(/r/n=1\*roman/r/ni/r/n)/r/n若/r/n時(shí)，/r/n，/r/n，僅當(dāng)/r/n時(shí)取等號，而/r/n，有/r/n，/r/n∴/r/n在/r/n上單調(diào)遞/r/n減/r/n，/r/n(/r/n=2\*roman/r/nii/r/n)/r/n若/r/n時(shí)，/r/n，/r/n，/r/n在/r/n上單調(diào)遞/r/n減/r/n，/r/n(/r/n=3\*roman/r/niii/r/n)/r/n若/r/n時(shí)，由/r/n得/r/n，/r/n，/r/n圖6/r/n且/r/n，有/r/n，/r/n與/r/n的/r/n圖6/r/n圖象如圖/r/n6/r/n所示，當(dāng)/r/n或/r/n時(shí)，/r/n，/r/n，/r/n在/r/n上單調(diào)遞/r/n減/r/n，/r/n當(dāng)/r/n時(shí)，/r/n，/r/n，/r/n在/r/n上單調(diào)遞增，/r/n綜上，當(dāng)/r/n時(shí)，/r/n的遞增區(qū)間為/r/n；當(dāng)/r/n時(shí)，/r/n的遞增區(qū)間為/r/n，遞減區(qū)間為/r/n；/r/n當(dāng)/r/n時(shí)，/r/n的遞減區(qū)間為/r/n./r/n評注：/r/n“/r/n雙圖法/r/n”/r/n可將抽象的問題直觀化，有效回避繁雜的不等式求解，且大大降低出錯(cuò)的機(jī)會/r/n./r/n2/r/n．/r/n直觀探/r/n求/r/n函數(shù)的/r/n極值/r/n/r/n運(yùn)用/r/n“/r/n雙圖法/r/n”/r/n求得/r/n的單調(diào)性后，/r/n再添/r/n加一條/r/n橫/r/n坐標(biāo)即可直觀地研究函數(shù)/r/n的極/r/n值/r/n與零點(diǎn)問題/r/n./r/n例/r/n2/r/n．/r/n已知函數(shù)/r/n，點(diǎn)/r/n./r/n若過點(diǎn)/r/n可作曲線/r/n的/r/n條切線，求/r/n的取值范圍/r/n./r/n解析：/r/n設(shè)過點(diǎn)/r/n的直線/r/n與曲線/r/n切于點(diǎn)/r/n，/r/n，/r/n則直線/r/n的斜率/r/n，/r/n又/r/n，/r/n則/r/n，/r/n=1\*GB3/r/n①/r/n圖/r/n7/r/n方程/r/n=1\*GB3/r/n①/r/n有/r/n個(gè)不/r/n相/r/n等的實(shí)根，令/r/n，有/r/n，/r/n圖/r/n7/r/n令/r/n得/r/n或/r/n，/r/n與/r/n的圖象如圖/r/n7/r/n所示，/r/n當(dāng)/r/n或/r/n時(shí)，/r/n，/r/n為增函數(shù)；當(dāng)/r/n時(shí)，/r/n，/r/n為減函數(shù)，/r/n圖/r/n8/r/n∴/r/n，/r/n，/r/n圖/r/n8/r/n(/r/n1/r/n)/r/n如圖/r/n8/r/n，當(dāng)/r/n或/r/n，/r/n圖/r/n9/r/n即/r/n或/r/n時(shí)，/r/n僅有一個(gè)零點(diǎn)，方程/r/n=1\*GB3/r/n①/r/n只有/r/n1/r/n個(gè)實(shí)/r/n根/r/n，/r/n圖/r/n9/r/n此時(shí)過點(diǎn)/r/n僅可作曲線/r/n的/r/n1/r/n條切線/r/n；/r/n(/r/n2/r/n)/r/n如圖/r/n9/r/n，當(dāng)/r/n或/r/n，/r/n即/r/n或/r/n時(shí)，/r/n有/r/n兩/r/n個(gè)零點(diǎn)，方程/r/n=1\*GB3/r/n①/r/n有/r/n兩/r/n個(gè)/r/n不等/r/n實(shí)/r/n根/r/n，/r/n圖/r/n10/r/n此時(shí)過點(diǎn)/r/n僅可作曲線/r/n的/r/n2/r/n條切線/r/n；/r/n圖/r/n10/r/n(/r/n3/r/n)/r/n如圖/r/n10/r/n，當(dāng)/r/n，即/r/n時(shí)，/r/n有三個(gè)零點(diǎn)，方程/r/n=1\*GB3/r/n①/r/n有三個(gè)不等實(shí)根，/r/n此時(shí)過點(diǎn)/r/n僅可作曲線/r/n的/r/n3/r/n條切線；/r/n綜上，當(dāng)/r/n時(shí)，/r/n的取值范圍是/r/n；當(dāng)/r/n時(shí)，/r/n的取值范圍是/r/n；當(dāng)/r/n時(shí)，/r/n的取值范圍是/r/n./r/n評注：/r/n運(yùn)用/r/n“/r/n雙圖法/r/n”/r/n描繪出函數(shù)/r/n的圖象/r/n，其極值、零點(diǎn)即/r/n可/r/n一目了然/r/n./r/n3/r/n．直觀研究函數(shù)的最值/r/n/r/n運(yùn)用/r/n“/r/n雙圖法/r/n”/r/n求得/r/n的單調(diào)性/r/n，/r/n可直觀地求得其最大值、最小值/r/n./r/n例/r/n3/r/n．/r/n當(dāng)/r/n時(shí)，求函數(shù)/r/n在/r/n上的最大值與最小值/r/n./r/n解析：/r/n∵/r/n，/r/n∴/r/n，/r/n,/r/n(/r/n1/r/n)/r/n當(dāng)/r/n時(shí)，由/r/n得/r/n，/r/n在/r/n上單調(diào)遞增，/r/n0/r/n圖/r/n11/r/n∴/r/n，/r/n；/r/n0/r/n圖/r/n11/r/n(/r/n2/r/n)/r/n當(dāng)/r/n時(shí)，令/r/n得/r/n，/r/n當(dāng)/r/n時(shí)，/r/n，/r/n為減函數(shù)，/r/n當(dāng)/r/n時(shí)，/r/n，/r/n為增函數(shù)，/r/n與/r/n的圖象如圖/r/n11/r/n所示，/r/n=1\*GB3/r/n①/r/n當(dāng)/r/n，即/r/n時(shí)，/r/n在/r/n上單調(diào)遞增，/r/n，/r/n；/r/n=2\*GB3/r/n②/r/n當(dāng)/r/n，即/r/n時(shí)，則/r/n，/r/n若/r/n，即/r/n，也即/r/n時(shí)，/r/n，/r/n若/r/n，即/r/n，也即/r/n時(shí)，/r/n，/r/n=3\*GB3/r/