函数y=x(1-x)²在[0,1]的最大值是

函数y=x(1-x)²在[0,1]的最大值是
函数y=x(1-x)²在[0,1]的最大值是

函数y=x(1-x)²在[0,1]的最大值是
y=x³-2x²+x
y'=3x²-4x+1=0
则x=1/3,x=1
00,递增
1/3所以x=1/3是极大值,也是最大值
所以x=1/3,y最大=4/27

f(x) =x(1-x)^2
= x-2x^2+x^3
f'(x) = 1-4x+3x^2 =0
(3x-1)(x-1) =0
x=1/3 or 1
f''(x) = -4+6x
f''(1/3) <0 (max)
f''(1) >0 ( min)
max f(x) = f(1/3)
= ...

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f(x) =x(1-x)^2
= x-2x^2+x^3
f'(x) = 1-4x+3x^2 =0
(3x-1)(x-1) =0
x=1/3 or 1
f''(x) = -4+6x
f''(1/3) <0 (max)
f''(1) >0 ( min)
max f(x) = f(1/3)
= (1/3)(1-1/3)^2 = 4/27

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