形如奇函數(shù)y=x.(ax^2+b)的圖像畫(huà)法及性質(zhì)詳解D5

函數(shù)y=eq\f(183x,323x2+153)的性質(zhì)及圖像畫(huà)法主要內(nèi)容：本文主要介紹函數(shù)y=eq\f(183x,323x2+153)的定義域、值域、單調(diào)性、奇偶性、凸凹性等性質(zhì)，并簡(jiǎn)要畫(huà)出函數(shù)的圖像示意圖。函數(shù)的定義域：∵分母323x2+153≥153＞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(lái)解析，分子分母同時(shí)除x有：y=eq\f(183x,323x2+153)=eq\f(183,323x+eq\f(153,x)),對(duì)于分母g(x)=323x+eq\f(153,x)有：(1)當(dāng)x＞0時(shí)，g(x)≥2eq\r(323x*eq\f(153,x))=102eq\r(19)，取等號(hào)時(shí)x=eq\f(3,19)\r(19)≈0.69，則函數(shù)增區(qū)間為(0，0.69)，減區(qū)間為[0.69,+∞);(2)當(dāng)x＜0時(shí)，g(x)≤-2eq\r(323x*eq\f(153,x))=-102eq\r(19)，取等號(hào)時(shí)x=-eq\f(3,19)\r(19)≈-0.69，則函數(shù)增區(qū)間為(-0.69,0)，減區(qū)間為(-∞,-0.69]?；蛘撸脤?dǎo)數(shù)知識(shí)求解有：y=eq\f(183x,323x2+153),eq\f(dy,dx)=eq\f(183*(323x2+153)-2*323*183x2,(323x2+153)2)=-eq\f(183(323x2-153),(323x2+153)2),令eq\f(dy,dx)=0,則:323x2-153=0,即323x2=153，求出：x=±eq\f(3,19)\r(19)≈±0.69，函數(shù)單調(diào)性為：(1)當(dāng)x∈(-∞，-0.69)∪(0.69，+∞)時(shí)，eq\f(dy,dx)≤0，函數(shù)y為減函數(shù)；(2)當(dāng)x∈[-0.69，0.69]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)=-183eq\f(323x2-153,(323x2+153)2),eq\f(d2y,dx2)=-183*eq\f(2*323x(323x2+153)2-(323x2-153)*4*323x(323x2+153),(323x2+153)?)=-183*eq\f(2*323x(323x2+153)-4*323x(323x2-153),(323x2+153)3)=2*323*183*eq\f(x(323x2-3*153),(323x2+153)3).令eq\f(d2y,dx2)=0，則x=0或者323x2-3*153=0，求出:x=±eq\f(3,19)eq\r(57)≈±1.19，函數(shù)y凸凹性為：(1)當(dāng)x∈[-1.19，0]∪(1.19，+∞)時(shí)，eq\f(d2y,dx2)>0，函數(shù)y為凹函數(shù)；(2)當(dāng)x∈(-∞,-1.19)∪(0,1.19]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(183x,323x2+153),所以:f(-x)=eq\f(183(-x),323(-x)2+153)=-eq\f(183x,323x2+153)=-f(x).函數(shù)f(x)為奇函數(shù)，圖像關(guān)于原點(diǎn)對(duì)稱。函數(shù)的極限：Lim(x→-∞)eq\f(183x,323x2+153)=0，Lim(x→+∞)eq\f(183x,323x2+153)=0，Lim(x→0+)eq\f(183x,323x2+153)=0，Lim(x→0-)eq\f(183x,323x2+153)=0.函數(shù)的特征點(diǎn)圖表：x-2.19-1.69-1.19-0.6974820.691.191.692.19183x-400.77-309.27-217.77-126.27126.27217.77309.27400.77323x2+1531702.141075.52610.40306.78306.78610.401075.521702.14y-0.24-0.29-0.36-0.4100.410.360.290.24函數(shù)的圖像示意圖：y=eq\f(183x,323x2+153)y(0.69,0.41)(1.19,0.36)(2.19,0.24)