函數(shù)y=eq\f(95x,401x2+225)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(95x,401x2+225)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401x2+225≥225＞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(95x,401x2+225)=eq\f(95,401x+eq\f(225,x)),對(duì)于分母g(x)=401x+eq\f(225,x)有：(1)當(dāng)x＞0時(shí)，g(x)≥2eq\r(401x*eq\f(225,x))=30eq\r(401)，取等號(hào)時(shí)x=eq\f(15,401)\r(401)≈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(401x*eq\f(225,x))=-30eq\r(401)，取等號(hào)時(shí)x=-eq\f(15,401)\r(401)≈-0.75，則函數(shù)增區(qū)間為(-0.75,0)，減區(qū)間為(-∞,-0.75]。或者，用導(dǎo)數(shù)知識(shí)求解有：y=eq\f(95x,401x2+225),eq\f(dy,dx)=eq\f(95*(401x2+225)-2*401*95x2,(401x2+225)2)=-eq\f(95(401x2-225),(401x2+225)2),令eq\f(dy,dx)=0,則:401x2-225=0,即401x2=225，求出：x=±eq\f(15,401)\r(401)≈±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)=-95eq\f(401x2-225,(401x2+225)2),eq\f(d2y,dx2)=-95*eq\f(2*401x(401x2+225)2-(401x2-225)*4*401x(401x2+225),(401x2+225)?)=-95*eq\f(2*401x(401x2+225)-4*401x(401x2-225),(401x2+225)3)=2*401*95*eq\f(x(401x2-3*225),(401x2+225)3).令eq\f(d2y,dx2)=0，則x=0或者401x2-3*225=0，求出:x=±eq\f(15,401)eq\r(1203)≈±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(95x,401x2+225),所以:f(-x)=eq\f(95(-x),401(-x)2+225)=-eq\f(95x,401x2+225)=-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(95x,401x2+225)=0，Lim(x→+∞)eq\f(95x,401x2+225)=0，Lim(x→0+)eq\f(95x,401x2+225)=0，Lim(x→0-)eq\f(95x,401x2+225)=0.函數(shù)的特征點(diǎn)圖表：x-2.40-1.85-1.30-0.7540.751.301.852.4095x-228.00-175.75-123.50-71.2538071.25123.50175.75228.00401x2+2252534.761597.42902.69450.566641450.56902.691597.422534.76y-0.09-0.11-0.14-0.1600.160.140.110.09函數(shù)的圖像示意圖：y=eq\f(95x,401x2+225)y(0.75,0.16)(1.30,0.14)(2.40,0.09)