形如奇函數(shù)y=x.(ax^2+b)的圖像畫法及性質(zhì)詳解D3

函數(shù)y=eq\f(151x,364x2+18)的性質(zhì)及圖像畫法主要內(nèi)容：本文主要介紹函數(shù)y=eq\f(151x,364x2+18)的定義域、值域、單調(diào)性、奇偶性、凸凹性等性質(zhì)，并簡要畫出函數(shù)的圖像示意圖。函數(shù)的定義域：∵分母364x2+18≥18＞0，即分母為正的實(shí)數(shù)，再取倒數(shù)函數(shù)有意義，∴函數(shù)的定義域?yàn)槿w實(shí)數(shù)，即：(-∞，+∞)。函數(shù)的單調(diào)性：可用基本不等式來解析，分子分母同時(shí)除x有：y=eq\f(151x,364x2+18)=eq\f(151,364x+eq\f(18,x)),對于分母g(x)=364x+eq\f(18,x)有：(1)當(dāng)x＞0時(shí)，g(x)≥2eq\r(364x*eq\f(18,x))=12eq\r(182)，取等號時(shí)x=eq\f(3,182)\r(182)≈0.22，則函數(shù)增區(qū)間為(0，0.22)，減區(qū)間為[0.22,+∞);(2)當(dāng)x＜0時(shí)，g(x)≤-2eq\r(364x*eq\f(18,x))=-12eq\r(182)，取等號時(shí)x=-eq\f(3,182)\r(182)≈-0.22，則函數(shù)增區(qū)間為(-0.22,0)，減區(qū)間為(-∞,-0.22]?；蛘?，用導(dǎo)數(shù)知識求解有：y=eq\f(151x,364x2+18),eq\f(dy,dx)=eq\f(151*(364x2+18)-2*364*151x2,(364x2+18)2)=-eq\f(151(364x2-18),(364x2+18)2),令eq\f(dy,dx)=0,則:364x2-18=0,即364x2=18，求出：x=±eq\f(3,182)\r(182)≈±0.22，函數(shù)單調(diào)性為：(1)當(dāng)x∈(-∞，-0.22)∪(0.22，+∞)時(shí)，eq\f(dy,dx)≤0，函數(shù)y為減函數(shù)；(2)當(dāng)x∈[-0.22，0.22]時(shí)，eq\f(dy,dx)>0，此時(shí)函數(shù)y為增函數(shù)。函數(shù)的凸凹性：eq\f(dy,dx)=-151eq\f(364x2-18,(364x2+18)2),eq\f(d2y,dx2)=-151*eq\f(2*364x(364x2+18)2-(364x2-18)*4*364x(364x2+18),(364x2+18)?)=-151*eq\f(2*364x(364x2+18)-4*364x(364x2-18),(364x2+18)3)=2*364*151*eq\f(x(364x2-3*18),(364x2+18)3).令eq\f(d2y,dx2)=0，則x=0或者364x2-3*18=0，求出:x=±eq\f(3,182)eq\r(546)≈±0.39，函數(shù)y凸凹性為：(1)當(dāng)x∈[-0.39，0]∪(0.39，+∞)時(shí)，eq\f(d2y,dx2)>0，函數(shù)y為凹函數(shù)；(2)當(dāng)x∈(-∞,-0.39)∪(0,0.39]時(shí)，eq\f(d2y,dx2)≤0，函數(shù)y為凸函數(shù)。函數(shù)的奇偶性：因?yàn)?f(x)=eq\f(151x,364x2+18),所以:f(-x)=eq\f(151(-x),364(-x)2+18)=-eq\f(151x,364x2+18)=-f(x).函數(shù)f(x)為奇函數(shù)，圖像關(guān)于原點(diǎn)對稱。函數(shù)的極限：Lim(x→-∞)eq\f(151x,364x2+18)=0，Lim(x→+∞)eq\f(151x,364x2+18)=0，Lim(x→0+)eq\f(151x,364x2+18)=0，Lim(x→0-)eq\f(151x,364x2+18)=0.函數(shù)的特征點(diǎn)圖表：x-0.73-0.56-0.39-0.2220.220.390.560.73151x-110.23-84.56-58.89-33.2230233.2258.8984.56110.23364x2+18211.98132.1573.3635.62147435.6273.36132.15211.98y-0.52-0.64-0.80-0.9300.930.800.640.52函數(shù)的圖像示意圖：y=eq\f(151x,364x2+18)y(0.22,0.93)(0.39,0.80)(0.73,0.52)