函數(shù)y=82x?-79x2+62的性質(zhì)及其圖像示意圖本文主要內(nèi)容：介紹函數(shù)y=82x?-79x2+62的定義域、值域、單調(diào)性、奇偶性、極限和凸凹性，并通過函數(shù)的導(dǎo)數(shù)知識(shí)計(jì)算函數(shù)的單調(diào)區(qū)間和凸凹區(qū)間。函數(shù)的定義域：根據(jù)函數(shù)的特征，函數(shù)自變量x可以取全體實(shí)數(shù)，即函數(shù)的定義域?yàn)椋海?∞，+∞）。函數(shù)的值域：因?yàn)閥=82x?-79x2+62，則：82x?-79x2+62-y=0,對(duì)x2的二次方程有解，則：判別式△=792-4*82(-y)≥0，即：4*82y≥4*82*62-792，解得y≥eq\f(14095,328)≈42.98故函數(shù)的值域?yàn)椋篬eq\f(14095,328),+∞)。函數(shù)的單調(diào)性：∵y=82x?-79x2+62,∴eq\f(dy,dx)=4*82x3-2*79x,令eq\f(dy,dx)=0,則：4*82x3-2*79x=0,x(164x2-79)=0,即x?=0，或者x2=eq\f(79,164),進(jìn)一步求出：x?=-eq\f(1,82)eq\r(3239),x?=0,x?=eq\f(1,82)eq\r(3239)≈0.69,則：(1)當(dāng)x∈(-∞,-eq\f(1,82)eq\r(3239)],(0,eq\f(1,82)eq\r(3239))時(shí)，eq\f(dy,dx)<0,則此時(shí)函數(shù)為減函數(shù)，該區(qū)間為減區(qū)間。(2)當(dāng)x∈[-eq\f(1,82)eq\r(3239),0],[eq\f(1,82)eq\r(3239),+∞)時(shí)，eq\f(dy,dx)>0,則此時(shí)函數(shù)為增函數(shù)，該區(qū)間為增區(qū)間,當(dāng)x0=±eq\f(1,82)eq\r(3239)時(shí)，y有最小值：ymin=f(±eq\f(1,82)eq\r(3239))=82*(±eq\f(1,82)eq\r(3239))?-79*(±eq\f(1,82)eq\r(3239))2+62=eq\f(14095,328).函數(shù)的奇偶性：∵f(x)=y=82x?-79x2+62,∴f(-x)=82*(-x)?-79*(-x)2+62=82x?-79x2+62=f(x).即函數(shù)f(x)為偶函數(shù)，圖像關(guān)于y軸對(duì)稱。函數(shù)的極限：eq\s(lim,x→0)82x?-79x2+62=62;eq\s(lim,x→-∞)82x?-79x2+62=+∞;eq\s(lim,x→+∞)82x?-79x2+62=+∞.函數(shù)的凸凹性∵eq\f(dy,dx)=328x3-158x,∴eq\f(d2y,dx2)=984x2-158,令eq\f(d2y,dx2)=0，則：x2=eq\f(79,492),即：x?=-eq\f(1,246)eq\r(9717),x?=eq\f(1,246)eq\r(9717)≈0.40;則：(1)當(dāng)x∈(-∞,-eq\f(1,246)eq\r(9717)),(eq\f(1,246)eq\r(9717),+∞)時(shí)，eq\f(d2y,dx2)>0,則此時(shí)函數(shù)為凹函數(shù)，該區(qū)間為凹區(qū)間。(2)當(dāng)x∈[-eq\f(1,246)eq\r(9717)，eq\f(1,246)eq\r(9717)]時(shí)，eq\f(d2y,dx2)<0,則此時(shí)函數(shù)為凸函數(shù)，該區(qū)間為凸區(qū)間。函數(shù)的五點(diǎn)圖x00.400.550.690.8382x?02.107.5018.5938.9279x2012.6423.9037.6154.42y6251.4645.6042.9846.50x-0.83-0.69-0.55-0.40082x?38.9218.597.502.10079x254.4237.6123.9012.640y46.5042.9845.6051.4662函數(shù)的示意圖：y=82x?-79x2+62y(0,62)(-0.40,51.46)(0.40,51.46)(-0.83,46.50)