函數(shù)y=eq\f(28x,378x2+249)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(28x,378x2+249)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378x2+249≥249＞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(28x,378x2+249)=eq\f(28,378x+eq\f(249,x)),對(duì)于分母g(x)=378x+eq\f(249,x)有：(1)當(dāng)x＞0時(shí)，g(x)≥2eq\r(378x*eq\f(249,x))=18eq\r(1162)，取等號(hào)時(shí)x=eq\f(1,42)\r(1162)≈0.81，則函數(shù)增區(qū)間為(0，0.81)，減區(qū)間為[0.81,+∞);(2)當(dāng)x＜0時(shí)，g(x)≤-2eq\r(378x*eq\f(249,x))=-18eq\r(1162)，取等號(hào)時(shí)x=-eq\f(1,42)\r(1162)≈-0.81，則函數(shù)增區(qū)間為(-0.81,0)，減區(qū)間為(-∞,-0.81]。或者，用導(dǎo)數(shù)知識(shí)求解有：y=eq\f(28x,378x2+249),eq\f(dy,dx)=eq\f(28*(378x2+249)-2*378*28x2,(378x2+249)2)=-eq\f(28(378x2-249),(378x2+249)2),令eq\f(dy,dx)=0,則:378x2-249=0,即378x2=249，求出：x=±eq\f(1,42)\r(1162)≈±0.81，函數(shù)單調(diào)性為：(1)當(dāng)x∈(-∞，-0.81)∪(0.81，+∞)時(shí)，eq\f(dy,dx)≤0，函數(shù)y為減函數(shù)；(2)當(dāng)x∈[-0.81，0.81]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)=-28eq\f(378x2-249,(378x2+249)2),eq\f(d2y,dx2)=-28*eq\f(2*378x(378x2+249)2-(378x2-249)*4*378x(378x2+249),(378x2+249)?)=-28*eq\f(2*378x(378x2+249)-4*378x(378x2-249),(378x2+249)3)=2*378*28*eq\f(x(378x2-3*249),(378x2+249)3).令eq\f(d2y,dx2)=0，則x=0或者378x2-3*249=0，求出:x=±eq\f(1,42)eq\r(3486)≈±1.41，函數(shù)y凸凹性為：(1)當(dāng)x∈[-1.41，0]∪(1.41，+∞)時(shí)，eq\f(d2y,dx2)>0，函數(shù)y為凹函數(shù)；(2)當(dāng)x∈(-∞,-1.41)∪(0,1.41]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(28x,378x2+249),所以:f(-x)=eq\f(28(-x),378(-x)2+249)=-eq\f(28x,378x2+249)=-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(28x,378x2+249)=0，Lim(x→+∞)eq\f(28x,378x2+249)=0，Lim(x→0+)eq\f(28x,378x2+249)=0，Lim(x→0-)eq\f(28x,378x2+249)=0.函數(shù)的特征點(diǎn)圖表：x-2.61-2.01-1.41-0.8174840.811.412.012.6128x-73.08-56.28-39.48-22.6820955222.6839.4856.2873.08378x2+2492823.971776.161000.50497.0121171877017497.011000.501776.162823.97y-0.03-0.03-0.04-0.0500.050.040.030.03函數(shù)的圖像示意圖：y=eq\f(28x,378x2+249)y(0.81,0.05)(1.41,0.04)(2.61,0.03)