第1个回答 2019-08-21
求下列函数在指定闭区间上的最大值和最小值
(1)f(x7)=2x^3-17x^2+42x-28
[1,5]
解析:∵f(x)=2x^3-17x^2+42x-28
令f’(x)=6x^2-34x+42=0==>3x^2-17x+21=0==>x1=(17-√37)/6,x2=(17+√37)/6
f’’(x)=12x-34==>f”(x1)<0,∴函数f(x)在x1处取极大值f(x1)≈4.1863
f”(x2)>0,∴函数f(x)在x2处取极小值f(x2)≈-4.1493
f(1)=-1,f(5)=7
∴函数f(x)在区间[1,5]上最大值为f(5)=7,最小值为f(x2)≈-4.1493
(2)g(x)=e^x(x^2-4x+3)[-3,2]
解析:∵g(x)=e^x(x^2-4x+3)
令g’(x)=e^x(x^2-4x+3)+
e^x(2x-4)=e^x(x^2-2x-1)=0==>x1=1-√2,
x2=1+√2
g’’(x)=e^x(x^2-2x-1)+
e^x(x-2)=e^x(x^2-x-3)
g’’(x1)<0,∴函数g(x)在x1处取极大值g(x1)≈3.1909
g”(x2)>0,∴函数g(x)在x2处取极小值g(x2)≈-9.2626
g(-3)=1.1949,g(2)=-7.3891
∴函数g(x)在区间[-3,2]上最大值为g(x1)=3.1909,最小值为g(2)≈-7.3891