函數(shù)y=(9x+4)²(9x+16)³的主要性質(zhì)及圖像示意圖

函數(shù)y=(9x+4)2(9x+16)3的主要性質(zhì)及其圖像示意圖主要內(nèi)容：通過函數(shù)的定義域、值域、單調(diào)性、凸凹性及極限的性質(zhì)，并通過函數(shù)導(dǎo)數(shù)知識求解函數(shù)的單調(diào)區(qū)間和凸凹區(qū)間，并簡要畫出函數(shù)y=(9x+4)2(9x+16)3示意圖的過程與步驟。※.函數(shù)定義域根據(jù)函y=(9x+4)2(9x+16)3特征，可知函數(shù)自變量x可以取全體實數(shù)，即函數(shù)的定義域為：（-∞，+∞）※.函數(shù)一階導(dǎo)數(shù)：※.函數(shù)乘積求導(dǎo)法。∵y=(9x+4)2(9x+16)3，∴y'=18(9x+4)(9x+16)3+(9x+4)2*27*(9x+16)2，=(9x+4)(9x+16)2(162x+288+243x+108),=(9x+4)(9x+16)2(405x+396)※.取對數(shù)求導(dǎo)法?！遹=(9x+4)2(9x+16)3，取導(dǎo)數(shù)有：∴l(xiāng)ny=ln(9x+4)2(9x+16)3,即：lny=2ln(9x+4)+3ln(9x+16),兩邊同時對x求導(dǎo)：eq\f(y',y)=eq\f(18,9x+4)+eq\f(27,9x+16),y'=y(eq\f(18,9x+4)+eq\f(27,9x+16)),y'=(9x+4)2(9x+16)3(eq\f(18,9x+4)+eq\f(27,9x+16)),y'=(9x+4)(9x+16)2[18(9x+16)+27(9x+4)],y'=(9x+4)(9x+16)2(405x+396).令y'=0，有9x+4=0，405x+396=0，即：x1=-eq\f(4,9)，x2=-eq\f(44,45).(1)當(dāng)x∈(-∞，-eq\f(44,45))，(-eq\f(4,9),+∞)時，eq\f(dy,dx)>0,此時函數(shù)為增函數(shù)。(2)當(dāng)x∈[-eq\f(44,45)，-eq\f(4,9)]時，eq\f(dy,dx)<0,此時函數(shù)為減函數(shù)。※.函數(shù)的凸凹性∵y'=(9x+4)(9x+16)2(405x+396)∴y''=9(9x+16)2(405x+396)+(9x+4)[18(9x+16)(405x+396)+405(9x+16)2]=9(9x+16)2(405x+396)+(9x+4)(9x+16)[18(405x+396)+405(9x+16)]=(9x+16)[9(9x+16)(405x+396)+18(9x+4)(405x+396)+405(9x+4)(9x+16)]=(9x+16){(405x+396)[9(9x+16)+18(9x+4)]+405(9x+4)(9x+16)}=(9x+16)(131220x2+256608x+111456)=324(9x+16)(405x2+792x+344).令y''=0，則9x+16=0，或405x2+792x+344=0，即：x3=-eq\f(16,9)≈-1.7,x4=eq\f(-44-6\r(6),45)≈-1.30,x5=eq\f(-44+6\r(6),45)≈-0.65,此時函數(shù)的凸凹性性及凸凹區(qū)間為：(1)當(dāng)x∈(-∞,-1.77),(-1.30,-0.651)時，y''<0,此時函數(shù)y為凸函數(shù)。(2)當(dāng)x∈[-1.77,-1.30],[-0.651,+∞)時，y''>0,此時函數(shù)y為凹函數(shù)。.函數(shù)的極限lim(x→-∞)(9x+4)2(9x+16)3=-∞；lim(x→+∞)(9x+4)2(9x+16)3=+∞；lim(x→0)(9x+4)2(9x+16)3=42*163。.函數(shù)的五點圖x-6.777-1.77-1.30-0.97-0.651-0.440.0555(9x+4)2324914459.898123.043.4618020.25(9x+16)3-9112077.342373.241042.417284492.1y-296048880463385993609090965※.函數(shù)的示意圖y=(9x+4)2(9x+16)3y(-0.97,8599)(-1.30,4633)(-0.651,3609)