函數(shù)y=(2x3-3)lnx的性質(zhì)及圖像示意圖主要內(nèi)容：本文主要解析函數(shù)y=(2x3-3)lnx的定義域、單調(diào)性、凸凹性等性質(zhì)，并通過(guò)微積分求解函數(shù)的單調(diào)區(qū)間，簡(jiǎn)要畫(huà)出函數(shù)的示意圖?！?函數(shù)的定義域根據(jù)函數(shù)特征，對(duì)函數(shù)y1=lnx定義域要求自變量為正數(shù)，所以本題函數(shù)的定義域?yàn)椋?0,+∞)?！?函數(shù)的單調(diào)性本處以導(dǎo)數(shù)知識(shí)來(lái)解析函數(shù)的單調(diào)性，具體步驟為：因?yàn)閥=(2x3-3)lnx，對(duì)x求導(dǎo)，所以eq\f(dy,dx)=3*2x2*lnx+(2x3-3)*eq\f(1,x)=eq\f(3*2x3*lnx+2x3-3,x);設(shè)g(x)=3*2x3*lnx+2x3-3有:g(1)=2-3<0，g(m)=3*2*m3*lnm+2*m3-3,其中m≥2.又因?yàn)?g'(x)=9*2x2*lnx+3*2x3*eq\f(1,x)+3*2x2=9*2x2*lnx+6*2x2>0,可知g(x)為增函數(shù)，則方程3*2x3*lnx+2x3-3=0在區(qū)間(1,2)上只有一個(gè)實(shí)數(shù)根，通過(guò)切線法求出該近似值為x0=1.073。則此時(shí)函數(shù)的單調(diào)性為：(1)當(dāng)x∈(0,1.073)時(shí)，eq\f(dy,dx)＜0，此時(shí)函數(shù)為減函數(shù)。(2)當(dāng)x∈[1.073,+∞)時(shí)，eq\f(dy,dx)≥0，此時(shí)函數(shù)為增函數(shù)。※.函數(shù)的凸凹性∵eq\f(dy,dx)=eq\f(3*2x3*lnx+2x3-3,x)，繼續(xù)求導(dǎo)有:∴eq\f(d2y,dx2)=eq\f((9*2x2lnx+3*2x3*eq\f(1,x)+3*2x2)*x-(3*2x3*lnx+2x3-3),x2),=eq\f(9*2x3lnx+3*2x3+3*2x3-3*2x3*lnx-2x3+3,x2),=eq\f(6*2x3lnx+5*2x3+3,x2),設(shè)k(x)=6*2x3lnx+5*2x3+3，則k'(x)=18*2x2lnx+21*2x2=3*2x2(6*lnx+7)。令k'(x)=0，則lnx=-eq\f(7,6)，即x=eeq\s\up10(-\f(7,6))，此時(shí)有：(1)當(dāng)x∈(0,eeq\s\up10(-\f(7,6)))時(shí)，k'(x)＜0，則函數(shù)k(x)為減函數(shù)；(2)當(dāng)x∈[eeq\s\up10(-\f(7,6)),+∞)時(shí)，k'(x)＞0，則函數(shù)k(x)為增函數(shù)。故當(dāng)x=eeq\s\up10(-\f(7,6))，則函數(shù)k(x)取最小值，即：k(x)=6*2x3lnx+5*2x3+3≥2*eeq\s\up10(-\f(7,2))*(6*lneeq\s\up10(-\f(7,6))+5)+3=3-2*eeq\s\up10(-\f(7,2))>0,所以eq\f(d2y,dx2)＞0，此時(shí)函數(shù)y為凹函數(shù)?！?函數(shù)的五點(diǎn)圖X0.2150.6441.0731.5021.9312x3-3-2.980-2.466-0.5293.77711.400lnx-1.537-0.4400.0700.4070.658y4.5801.085-0.0371.5377.501※.函數(shù)的示意圖y=(2x3-3)lnx(1.931,7.501)(0.215,4.580)