函數(shù)y=57x?-12x2+28的性質(zhì)及其圖像示意圖本文主要內(nèi)容：介紹函數(shù)y=57x?-12x2+28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=57x?-12x2+28，則：57x?-12x2+28-y=0,對x2的二次方程有解，則：判別式△=122-4*57(-y)≥0，即：4*57y≥4*57*28-122，解得y≥eq\f(520,19)≈27.37故函數(shù)的值域?yàn)椋篬eq\f(520,19),+∞)。函數(shù)的單調(diào)性：∵y=57x?-12x2+28,∴eq\f(dy,dx)=4*57x3-2*12x,令eq\f(dy,dx)=0,則：4*57x3-2*12x=0,x(114x2-12)=0,即x?=0，或者x2=eq\f(2,19),進(jìn)一步求出：x?=-eq\f(1,19)eq\r(38),x?=0,x?=eq\f(1,19)eq\r(38)≈0.32,則：(1)當(dāng)x∈(-∞,-eq\f(1,19)eq\r(38)],(0,eq\f(1,19)eq\r(38))時，eq\f(dy,dx)<0,則此時函數(shù)為減函數(shù)，該區(qū)間為減區(qū)間。(2)當(dāng)x∈[-eq\f(1,19)eq\r(38),0],[eq\f(1,19)eq\r(38),+∞)時，eq\f(dy,dx)>0,則此時函數(shù)為增函數(shù)，該區(qū)間為增區(qū)間,當(dāng)x0=±eq\f(1,19)eq\r(38)時，y有最小值：ymin=f(±eq\f(1,19)eq\r(38))=57*(±eq\f(1,19)eq\r(38))?-12*(±eq\f(1,19)eq\r(38))2+28=eq\f(520,19).函數(shù)的奇偶性：∵f(x)=y=57x?-12x2+28,∴f(-x)=57*(-x)?-12*(-x)2+28=57x?-12x2+28=f(x).即函數(shù)f(x)為偶函數(shù)，圖像關(guān)于y軸對稱。函數(shù)的極限：eq\s(lim,x→0)57x?-12x2+28=28;eq\s(lim,x→-∞)57x?-12x2+28=+∞;eq\s(lim,x→+∞)57x?-12x2+28=+∞.函數(shù)的凸凹性∵eq\f(dy,dx)=228x3-24x,∴eq\f(d2y,dx2)=684x2-24,令eq\f(d2y,dx2)=0，則：x2=eq\f(2,57),即：x?=-eq\f(1,57)eq\r(114),x?=eq\f(1,57)eq\r(114)≈0.19;則：(1)當(dāng)x∈(-∞,-eq\f(1,57)eq\r(114)),(eq\f(1,57)eq\r(114),+∞)時，eq\f(d2y,dx2)>0,則此時函數(shù)為凹函數(shù)，該區(qū)間為凹區(qū)間。(2)當(dāng)x∈[-eq\f(1,57)eq\r(114)，eq\f(1,57)eq\r(114)]時，eq\f(d2y,dx2)<0,則此時函數(shù)為凸函數(shù)，該區(qū)間為凸區(qū)間。函數(shù)的五點(diǎn)圖x00.190.260.320.3857x?00.070.260.601.1912x200.430.811.231.73y2827.6427.4527.3727.46x-0.38-0.32-0.26-0.19057x?1.190.600.260.07012x21.731.230.810.430y27.4627.3727.4527.6428函數(shù)的示意圖：y=57x?-12x2+28y(0,28)(-0.19,27.64)(0.19,27.64)(-0.38,27.46)