函數(shù)y=eq\f(13,(30x+38)3)的主要性質(zhì)與圖像主要內(nèi)容：本題主要介紹函數(shù)y=eq\f(13,(30x+38)3)的定義域、值域、單調(diào)性、凸凹性、極限等性質(zhì)，并通過(guò)函數(shù)導(dǎo)數(shù)知識(shí)求解函數(shù)的單調(diào)區(qū)間和凸凹區(qū)間。函數(shù)的定義域：該函數(shù)y=eq\f(13,(30x+38)3)為分式函數(shù)，要求分母不為0，因?yàn)?0x+38≠0，則x≠-eq\f(19,15),故函數(shù)的定義域?yàn)椋?-∞,-eq\f(19,15))，(-eq\f(19,15),+∞)。函數(shù)導(dǎo)數(shù)法解析單調(diào)性：因?yàn)閥=eq\f(13,(30x+38)3)，對(duì)x求導(dǎo)，所以有：eq\f(dy,dx)=-3*390*eq\f((30x+38)2,(30x+38)6)=-eq\f(1170,(30x+38)4)＜0，所以函數(shù)y=eq\f(13,(30x+38)3)在定義域上為單調(diào)減函數(shù)。函數(shù)的凸凹性：由eq\f(dy,dx)=-eq\f(1170,(30x+38)4)得：eq\f(d2y,dx2)=-1170*(30x+38)-4，再次對(duì)x求導(dǎo)，有：eq\f(d2y,dx2)=-1170*(-4)*(30x+38)-5*30=140400(30x+38)-5，該二階導(dǎo)數(shù)的間斷點(diǎn)為x=-eq\f(19,15)，此時(shí)函數(shù)的凸凹性及凸凹區(qū)間為：(1)當(dāng)x∈(-∞,-eq\f(19,15))時(shí)，分母<0，則eq\f(d2y,dx2)<0,此時(shí)函數(shù)y為單調(diào)凸函數(shù)，(2)當(dāng)x∈(-eq\f(19,15),+∞)時(shí)，分母>0，則eq\f(d2y,dx2)>0,此時(shí)函數(shù)y為單調(diào)凹函數(shù)。函數(shù)的極限：lim(x→-∞)eq\f(13,(30x+38)3)=0；lim(x+→-eq\f(19,15))eq\f(13,(30x+38)3)=+∞；lim(x-→-eq\f(19,15))eq\f(13,(30x+38)3)=+∞；lim(x→+∞)eq\f(13,(30x+38)3)=0。函數(shù)的五點(diǎn)圖x-3.8-3.3-2.8-2.3-1.830x+38-76-61-46-31-16y-0.00003-0.00006-0.00013-0.00044-0.00317x-0.8-0.30.20.71.230x+381429445974y0.004730.000530.000150.000060.00003函數(shù)的示意圖：y=13/(30x+38)^3x=-19/15y(-0.8,0.00473)(-0.3,0.00053)(-3.8,-0.00003)X(-2.3,-0.00044)(1.2,0.00003)