形如奇函數(shù)y=x.(ax^2+b)的圖像畫(huà)法及性質(zhì)詳解D9

函數(shù)y=eq\f(114x,89x2+83)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(114x,89x2+83)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89x2+83≥83＞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(114x,89x2+83)=eq\f(114,89x+eq\f(83,x)),對(duì)于分母g(x)=89x+eq\f(83,x)有：(1)當(dāng)x＞0時(shí)，g(x)≥2eq\r(89x*eq\f(83,x))=2eq\r(7387)，取等號(hào)時(shí)x=eq\f(1,89)\r(7387)≈0.97，則函數(shù)增區(qū)間為(0，0.97)，減區(qū)間為[0.97,+∞);(2)當(dāng)x＜0時(shí)，g(x)≤-2eq\r(89x*eq\f(83,x))=-2eq\r(7387)，取等號(hào)時(shí)x=-eq\f(1,89)\r(7387)≈-0.97，則函數(shù)增區(qū)間為(-0.97,0)，減區(qū)間為(-∞,-0.97]。或者，用導(dǎo)數(shù)知識(shí)求解有：y=eq\f(114x,89x2+83),eq\f(dy,dx)=eq\f(114*(89x2+83)-2*89*114x2,(89x2+83)2)=-eq\f(114(89x2-83),(89x2+83)2),令eq\f(dy,dx)=0,則:89x2-83=0,即89x2=83，求出：x=±eq\f(1,89)\r(7387)≈±0.97，函數(shù)單調(diào)性為：(1)當(dāng)x∈(-∞，-0.97)∪(0.97，+∞)時(shí)，eq\f(dy,dx)≤0，函數(shù)y為減函數(shù)；(2)當(dāng)x∈[-0.97，0.97]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)=-114eq\f(89x2-83,(89x2+83)2),eq\f(d2y,dx2)=-114*eq\f(2*89x(89x2+83)2-(89x2-83)*4*89x(89x2+83),(89x2+83)?)=-114*eq\f(2*89x(89x2+83)-4*89x(89x2-83),(89x2+83)3)=2*89*114*eq\f(x(89x2-3*83),(89x2+83)3).令eq\f(d2y,dx2)=0，則x=0或者89x2-3*83=0，求出:x=±eq\f(1,89)eq\r(22161)≈±1.67，函數(shù)y凸凹性為：(1)當(dāng)x∈[-1.67，0]∪(1.67，+∞)時(shí)，eq\f(d2y,dx2)>0，函數(shù)y為凹函數(shù)；(2)當(dāng)x∈(-∞,-1.67)∪(0,1.67]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(114x,89x2+83),所以:f(-x)=eq\f(114(-x),89(-x)2+83)=-eq\f(114x,89x2+83)=-f(x).函數(shù)f(x)為奇函數(shù)，圖像關(guān)于原點(diǎn)對(duì)稱(chēng)。函數(shù)的極限：Lim(x→-∞)eq\f(114x,89x2+83)=0，Lim(x→+∞)eq\f(114x,89x2+83)=0，Lim(x→0+)eq\f(114x,89x2+83)=0，Lim(x→0-)eq\f(114x,89x2+83)=0.函數(shù)的特征點(diǎn)圖表：x-3.07-2.37-1.67-0.9760.971.672.373.07114x-349.98-270.18-190.38-110.58684110.58190.38270.18349.9889x2+83921.82582.90331.21166.743287166.74331.21582.90921.82y-0.38-0.46-0.57-0.6600.660.570.460.38函數(shù)的圖像示意圖：y=eq\f(114x,89x2+83)y(0.97,0.66)(1.67,0.57)(3.07,0.38)