函數(shù)y=(x+1)2(x+2)3的主要性質(zhì)及圖像示意圖的畫(huà)法步驟

解析函數(shù)y=(x+1)2(x+2)3的主要性質(zhì)主要內(nèi)容：通過(guò)函數(shù)的定義域、值域、單調(diào)性、凸凹性及極限的性質(zhì)，并通過(guò)函數(shù)導(dǎo)數(shù)知識(shí)求解函數(shù)的單調(diào)區(qū)間和凸凹區(qū)間，并簡(jiǎn)要畫(huà)出函數(shù)y=(x+1)2(x+2)3示意圖的過(guò)程與步驟?！?函數(shù)定義域根據(jù)函y=(x+1)2(x+2)3特征，可知函數(shù)自變量x可以取全體實(shí)數(shù)，即函數(shù)的定義域?yàn)椋海?∞，+∞）※.函數(shù)一階導(dǎo)數(shù)：※.函數(shù)乘積求導(dǎo)法?！遹=(x+1)2(x+2)3，∴y'=2(x+1)(x+2)3+(x+1)2*3*(x+2)2，=(x+1)(x+2)2(2x+4+3x+3),=(x+1)(x+2)2(5x+7)※.取對(duì)數(shù)求導(dǎo)法。∵y=(x+1)2(x+2)3，取導(dǎo)數(shù)有：∴l(xiāng)ny=ln(x+1)2(x+2)3,即：lny=2ln(x+1)+3ln(x+2),兩邊同時(shí)對(duì)x求導(dǎo)：eq\f(y',y)=eq\f(2,x+1)+eq\f(3,x+2),y'=y(eq\f(2,x+1)+eq\f(3,x+2)),y'=(x+1)2(x+2)3(eq\f(2,x+1)+eq\f(3,x+2)),y'=(x+1)(x+2)2[2(x+2)+3(x+1)],y'=(x+1)(x+2)2(5x+7).令y'=0，有x+1=0，5x+7=0，即：x1=-1，x2=-eq\f(7,5).(1)當(dāng)x∈(-∞，-eq\f(7,5))，(-1,+∞)時(shí)，eq\f(dy,dx)>0,此時(shí)函數(shù)為增函數(shù)。(2)當(dāng)x∈[-eq\f(7,5)，-1]時(shí)，eq\f(dy,dx)<0,此時(shí)函數(shù)為減函數(shù)?！?函數(shù)的凸凹性∵y'=(x+1)(x+2)2(5x+7)∴y''=(x+2)2(5x+7)+(x+1)[2(x+2)(5x+7)+5(x+2)2]=(x+2)2(5x+7)+(x+1)(x+2)[2(5x+7)+5(x+2)]=(x+2)[(x+2)(5x+7)+2(x+1)(5x+7)+5(x+1)(x+2)]=(x+2){(5x+7)[(x+2)+2(x+1)]+5(x+1)(x+2)}=(x+2)(20x2+56x+38)=2(x+2)(10x2+28x+19).令y''=0，則x+2=0，或10x2+28x+19=0，即：x3=-2,x4=eq\f(-14-\r(6),10)≈-1.64,x5=eq\f(-14+\r(6),10)≈-1.15,此時(shí)函數(shù)的凸凹性性及凸凹區(qū)間為：(1)當(dāng)x∈(-∞,-2),(-1.64,-1.155)時(shí)，y''<0,此時(shí)函數(shù)y為凸函數(shù)。(2)當(dāng)x∈[-2,-1.64],[-1.155,+∞)時(shí)，y''>0,此時(shí)函數(shù)y為凹函數(shù)。.函數(shù)的極限lim(x→-∞)(x+1)2(x+2)3=-∞；lim(x→+∞)(x+1)2(x+2)3=+∞；lim(x→0)(x+1)2(x+2)3=12*23。.函數(shù)的五點(diǎn)圖x-3-2-1.64-1.4-1.16-1-0.5(x+1)2410.420.160.0200.25(x+2)3-100.040.220.6013.38y-400.020.030.0100.84※.函數(shù)的示意圖 y(-1.4,0.03)(-1.64,0.02)(-1.155,0.01)