Question

4x³—4x²+x的最大值

求出下列函数在指定区间上的最大值和最小值
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