求出下列函数在指定区间上的最大值和最小值

Question

F(x)=2x^3+x^2-4x+1 [-2,1]
G(x)=(e^x)(x^2-4x+3) [-3,2]

Answer 1

F(x) = 2x³ + x² - 4x + 1，x∈[-2，1]
F'(x) = 6x² + 2x - 4
F''(x) = 12x + 2
F'(x) = 0 => x = -1 OR x = 2/3
F''(-1) < 0，取得极大值；F''(2/3) > 0，取得最小值
F(-1) = 4，F(2/3) = -17/27
F(-2) = -3，F(1) = 0
∴最小值 = -3，最大值 = 4
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G(x) = (x² - 4x + 3)e^x，x∈[-3，2]
G'(x) = (2x - 4)e^x + (x² - 4x + 3)e^x
= 2xe^x - 4e^x + x²e^x - 4xe^x + 3e^x
= x²e^x - 2xe^x - e^x
= (x² - 2x - 1)e^x
G''(x) = (2x - 2)e^x + (x² - 2x - 1)e^x
= 2xe^x - 2e^x + x²e^x - 2xe^x - e^x
= x²e^x - 3e^x
= (x² - 3)e^x
G'(x) = 0 => x = 1 - √2 OR x = 1 + √2
G''(1 - √2) < 0，取得极大值；G''(1 + √2) > 0，取得极小值
G(1 - √2) = 2(1 + √2)e^(1 - √2) ≈ 3.19
G(1 + √2) = 2(1 - √2)e^(1 + √2) ≈ -9.26
G(-3) = 24/e³ ≈ 1.19
G(2) = -e² ≈ -7.39
最小值 = 2(1 - √2)e^(1 + √2) ≈ -9.26，最大值 = 2(1 + √2)e^(1 - √2) ≈ 3.19

Answer 2

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