函數(shù)y=eq\f(292x,292x2+165)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(292x,292x2+165)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292x2+165≥165＞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(292x,292x2+165)=eq\f(292,292x+eq\f(165,x)),對(duì)于分母g(x)=292x+eq\f(165,x)有：(1)當(dāng)x＞0時(shí)，g(x)≥2eq\r(292x*eq\f(165,x))=4eq\r(12045)，取等號(hào)時(shí)x=eq\f(1,146)\r(12045)≈0.75，則函數(shù)增區(qū)間為(0，0.75)，減區(qū)間為[0.75,+∞);(2)當(dāng)x＜0時(shí)，g(x)≤-2eq\r(292x*eq\f(165,x))=-4eq\r(12045)，取等號(hào)時(shí)x=-eq\f(1,146)\r(12045)≈-0.75，則函數(shù)增區(qū)間為(-0.75,0)，減區(qū)間為(-∞,-0.75]?；蛘撸脤?dǎo)數(shù)知識(shí)求解有：y=eq\f(292x,292x2+165),eq\f(dy,dx)=eq\f(292*(292x2+165)-2*292*292x2,(292x2+165)2)=-eq\f(292(292x2-165),(292x2+165)2),令eq\f(dy,dx)=0,則:292x2-165=0,即292x2=165，求出：x=±eq\f(1,146)\r(12045)≈±0.75，函數(shù)單調(diào)性為：(1)當(dāng)x∈(-∞，-0.75)∪(0.75，+∞)時(shí)，eq\f(dy,dx)≤0，函數(shù)y為減函數(shù)；(2)當(dāng)x∈[-0.75，0.75]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)=-292eq\f(292x2-165,(292x2+165)2),eq\f(d2y,dx2)=-292*eq\f(2*292x(292x2+165)2-(292x2-165)*4*292x(292x2+165),(292x2+165)?)=-292*eq\f(2*292x(292x2+165)-4*292x(292x2-165),(292x2+165)3)=2*292*292*eq\f(x(292x2-3*165),(292x2+165)3).令eq\f(d2y,dx2)=0，則x=0或者292x2-3*165=0，求出:x=±eq\f(3,146)eq\r(4015)≈±1.30，函數(shù)y凸凹性為：(1)當(dāng)x∈[-1.30，0]∪(1.30，+∞)時(shí)，eq\f(d2y,dx2)>0，函數(shù)y為凹函數(shù)；(2)當(dāng)x∈(-∞,-1.30)∪(0,1.30]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(292x,292x2+165),所以:f(-x)=eq\f(292(-x),292(-x)2+165)=-eq\f(292x,292x2+165)=-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(292x,292x2+165)=0，Lim(x→+∞)eq\f(292x,292x2+165)=0，Lim(x→0+)eq\f(292x,292x2+165)=0，Lim(x→0-)eq\f(292x,292x2+165)=0.函數(shù)的特征點(diǎn)圖表：x-2.40-1.85-1.30-0.7500.751.301.852.40292x-700.80-540.20-379.60-219.000219.00379.60540.20700.80292x2+1651846.921164.37658.48329.25165329.25658.481164.371846.92y-0.38-0.46-0.58-0.6700.670.580.460.38函數(shù)的圖像示意圖：y=eq\f(292x,292x2+165)y(0.75,0.67)(1.30,0.58)(2.40,0.38)