函數(shù)y=根號x(31x+29除x)的圖像示意圖及主要性質(zhì)

函數(shù)y=eq\r(x)(31x+eq\f(29,x))的圖像示意圖及主要性質(zhì)主要內(nèi)容：本文主要介紹根式分式復(fù)合函數(shù)y=eq\r(x)(31x+eq\f(29,x))的定義域、值域、單調(diào)和凸凹性等性質(zhì)，并通過導(dǎo)數(shù)知識計算函數(shù)的單調(diào)區(qū)間和凸凹區(qū)間，畫出y=eq\r(x)(31x+eq\f(29,x))的圖像示意圖?！?函數(shù)的定義域∵eq\r(x)有x≥0；對eq\f(29,x)有x≠0.∴函數(shù)的定義域為：(0，+∞）?！?函數(shù)的單調(diào)性∵y=eq\r(x)(31x+eq\f(29,x))=31xeq\s\up15(\f(3,2))+29xeq\s\up15(-\f(1,2)),對x求導(dǎo)得：∴eq\f(dy,dx)=eq\f(3,2)*31xeq\s\up15(\f(1,2))-eq\f(29,2)xeq\s\up15(-\f(3,2))=eq\f(1,2)xeq\s\up15(-\f(3,2))*(3*31x2-29).令eq\f(dy,dx)=0,則x2=eq\f(29,93).又因為x>0，則x=eq\f(1,93)eq\r(2697)≈0.56.(1)當(dāng)x∈(0,0.56)時，eq\f(dy,dx)＜0,函數(shù)y為單調(diào)減函數(shù)；(2)當(dāng)x∈[0.56,+∞)時，eq\f(dy,dx)＞0,函數(shù)y為單調(diào)增函數(shù)。※.函數(shù)的凸凹性∵eq\f(dy,dx)=eq\f(1,2)xeq\s\up15(-\f(3,2))*(3*31x2-29)，∴eq\f(d2y,dx2)=-eq\f(3,4)*xeq\s\up15(-\f(5,2))(3*31x2-29)+3*31x*xeq\s\up15(-\f(3,2)),=-eq\f(3,4)*xeq\s\up15(-\f(5,2))(3*31x2-29)+3*31xeq\s\up15(-\f(1,2)),=-eq\f(3,4)xeq\s\up15(-\f(5,2))(3*31x2-29-4*31x2),=eq\f(3,4)xeq\s\up15(-\f(5,2))(31x2+29)>0,則：函數(shù)y在定義域上為凹函數(shù)?！?函數(shù)的極限lim(x→0)eq\r(x)(31x+eq\f(29,x))=+∞lim(x→+∞)eq\r(x)(31x+eq\f(29,x))=+∞?！?函數(shù)的五點圖x0.110.340.560.781.00eq\r(x)0.330.580.750.881.0031x+eq\f(29,x)267.0595.8369.1561.3660.00y88.1355.5851.8654.0060.00※.函數(shù)的圖像y=eq\r(x)(31x+eq\f(29,x))y(0.11，88.13）