f(x)=2x^3-3ax^2+2(x∈R,a>0)(1)若f(x)在点(1,f(1))处的切线与Y=-1/3x+1垂直
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f(x)=2x^3-3ax^2+2(x∈R,a>0)(1)若f(x)在点(1,f(1))处的切线与Y=-1/3x+1垂直,求f(x)的单调区间.
(2)试求f(x)在[0,2]上的最大值.
(2)试求f(x)在[0,2]上的最大值.
(1) f'(x) = 6x^2 - 6ax
f'(1) = 6 -6a
y = -x/3 + 1的斜率k = -1/3
f(x)在点(1,f(1))处的切线与y= -x/3 +1垂直,此切线的斜率= -(-1/3) = 3
f'(1) = 6 -6a = 3
a = 1/2
f(x) = 2x^3 -3x^2/2 +2
f'(x) = 6x^2 - 3x = 3x(2x -1)
x < 0:2x - 1 < 0,x(2x -1) > 0,f(x)递增
0 < x < 1/2:x > 0,2x -1 < 0,x(2x -1) > 0,f(x)递减
x > 1/2:x > 0:2x - 1 > 0,x(2x -1) > 0,f(x)递增
再问: 第一问会,主要是第二问,我算的是两种情况,a>4/3时,最大值是f(2)=18-2a.和a∈(0,4/3]时,最大值是f(0)=2,对么?
再答: (2) f'(x) = 6x^2 - 6ax = 6x(x - a) = 0 x = 0, x = a (a > 0) x < 0: x -a < 0, x(x -a) > 0, f(x)递增 0 < x < a: x > 0, x - a < 0, x(x - a) > 0, f(x)递减 x > a: x > 0: x - a > 0, x(x - a) > 0, f(x)递增 a > 2: f(x)在(0,2)上递减, 最大值是f(0) = 2 0 < a < 2: f(x)在(0,a)上递减, 在(a,2)上递增; 最大值是f(0)和f(2)的较大者 f(0) = 2, f(2) = 18 - 12a (A) f(0) > f(2): 2 > 18 - 12a, a > 4/3 即4/3 < a < 2时, 最大值是f(0) = 2 (B) f(0) < f(2): 2 < 18 - 12a, a < 4/3 即 0 < a < 4/3时, 最大值是f(2) = 18-12a (C) a = 4/3: f(0)=f(2), 都是最大值
f'(1) = 6 -6a
y = -x/3 + 1的斜率k = -1/3
f(x)在点(1,f(1))处的切线与y= -x/3 +1垂直,此切线的斜率= -(-1/3) = 3
f'(1) = 6 -6a = 3
a = 1/2
f(x) = 2x^3 -3x^2/2 +2
f'(x) = 6x^2 - 3x = 3x(2x -1)
x < 0:2x - 1 < 0,x(2x -1) > 0,f(x)递增
0 < x < 1/2:x > 0,2x -1 < 0,x(2x -1) > 0,f(x)递减
x > 1/2:x > 0:2x - 1 > 0,x(2x -1) > 0,f(x)递增
再问: 第一问会,主要是第二问,我算的是两种情况,a>4/3时,最大值是f(2)=18-2a.和a∈(0,4/3]时,最大值是f(0)=2,对么?
再答: (2) f'(x) = 6x^2 - 6ax = 6x(x - a) = 0 x = 0, x = a (a > 0) x < 0: x -a < 0, x(x -a) > 0, f(x)递增 0 < x < a: x > 0, x - a < 0, x(x - a) > 0, f(x)递减 x > a: x > 0: x - a > 0, x(x - a) > 0, f(x)递增 a > 2: f(x)在(0,2)上递减, 最大值是f(0) = 2 0 < a < 2: f(x)在(0,a)上递减, 在(a,2)上递增; 最大值是f(0)和f(2)的较大者 f(0) = 2, f(2) = 18 - 12a (A) f(0) > f(2): 2 > 18 - 12a, a > 4/3 即4/3 < a < 2时, 最大值是f(0) = 2 (B) f(0) < f(2): 2 < 18 - 12a, a < 4/3 即 0 < a < 4/3时, 最大值是f(2) = 18-12a (C) a = 4/3: f(0)=f(2), 都是最大值
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