求下列函式在指定閉區間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)

F”(x2)>0，∴函式F(x)在x2處取極小值F(x2)≈-4.1493

F(1)=-1，F(5)=7

∴函式F(x)在區間[1,5]上最大值為F(5)=7，最小值為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)

G”(x2)>0，∴函式G(x)在x2處取極小值G(x2)≈-9.2626

G(-3)=1.1949，G(2)=-7.3891

∴函式G(x)在區間[-3,2]上最大值為G(x1)=3.1909，最小值為G(2)≈-7.3891