函數(shù)y=37x?-28x2+22的性質及其圖像示意圖本文主要內容：介紹函數(shù)y=37x?-28x2+22的定義域、值域、單調性、奇偶性、極限和凸凹性，并通過函數(shù)的導數(shù)知識計算函數(shù)的單調區(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可以取全體實數(shù)，即函數(shù)的定義域為：（-∞，+∞）。函數(shù)的值域：因為y=37x?-28x2+22，則：37x?-28x2+22-y=0,對x2的二次方程有解，則：判別式△=282-4*37(-y)≥0，即：4*37y≥4*37*22-282，解得y≥eq\f(618,37)≈16.71故函數(shù)的值域為：[eq\f(618,37),+∞)。函數(shù)的單調性：∵y=37x?-28x2+22,∴eq\f(dy,dx)=4*37x3-2*28x,令eq\f(dy,dx)=0,則：4*37x3-2*28x=0,x(74x2-28)=0,即x?=0，或者x2=eq\f(14,37),進一步求出：x?=-eq\f(1,37)eq\r(518),x?=0,x?=eq\f(1,37)eq\r(518)≈0.62,則：(1)當x∈(-∞,-eq\f(1,37)eq\r(518)],(0,eq\f(1,37)eq\r(518))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x∈[-eq\f(1,37)eq\r(518),0],[eq\f(1,37)eq\r(518),+∞)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x0=±eq\f(1,37)eq\r(518)時，y有最小值：ymin=f(±eq\f(1,37)eq\r(518))=37*(±eq\f(1,37)eq\r(518))?-28*(±eq\f(1,37)eq\r(518))2+22=eq\f(618,37).函數(shù)的奇偶性：∵f(x)=y=37x?-28x2+22,∴f(-x)=37*(-x)?-28*(-x)2+22=37x?-28x2+22=f(x).即函數(shù)f(x)為偶函數(shù)，圖像關于y軸對稱。函數(shù)的極限：eq\s(lim,x→0)37x?-28x2+22=22;eq\s(lim,x→-∞)37x?-28x2+22=+∞;eq\s(lim,x→+∞)37x?-28x2+22=+∞.函數(shù)的凸凹性∵eq\f(dy,dx)=148x3-56x,∴eq\f(d2y,dx2)=444x2-56,令eq\f(d2y,dx2)=0，則：x2=eq\f(14,111),即：x?=-eq\f(1,111)eq\r(1554),x?=eq\f(1,111)eq\r(1554)≈0.36;則：(1)當x∈(-∞,-eq\f(1,111)eq\r(1554)),(eq\f(1,111)eq\r(1554),+∞)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x∈[-eq\f(1,111)eq\r(1554)，eq\f(1,111)eq\r(1554)]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)的五點圖x00.360.490.620.7537x?00.622.135.4711.7128x203.636.7210.7615.75y2218.9917.4116.7117.96x-0.75-0.62-0.49-0.36037x?11.715.472.130.62028x215.7510.766.723.630y17.9616.7117.4118.9922函數(shù)的示意圖：y=37x?-28x2+22y(0,22)(-0.36,18.99)(0.36,18.99)(-0.75,17.96)