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受欢迎的函数 >

extreme f(x)=x^2(2-4x)^3

解答

极值点

f (x) = x^{2} (2 - 4 x)^{3}

解答

极小值 (0, 0), 极大值 (\frac{1}{5}, \frac{216}{3125}), 鞍点 (\frac{1}{2}, 0)

求解步骤

f^{'} (x) = - 16 x (2 x - 1)^{2} (5 x - 1)

Find intervals:

递减

: - \infty < x < 0,

递增

: 0 < x < \frac{1}{5},

递减

: \frac{1}{5} < x < \frac{1}{2},

递减

: \frac{1}{2} < x < \infty

将

x = 0

代入

x^{2} (2 - 4 x)^{3} : 0

极小值 (0, 0)

将

x = \frac{1}{5}

代入

x^{2} (2 - 4 x)^{3} : \frac{216}{3125}

极大值 (\frac{1}{5}, \frac{216}{3125})

将

x = \frac{1}{2}

代入

x^{2} (2 - 4 x)^{3} : 0

鞍点 (\frac{1}{2}, 0)

极小值 (0, 0), 极大值 (\frac{1}{5}, \frac{216}{3125}), 鞍点 (\frac{1}{2}, 0)

作图

流行的例子

extreme f(x,y)=e^{-(x^2+y^2)}(x+y^2)+(1/2)

e x t re m e f (x, y) = e^{- (x^{2} + y^{2})} (x + y^{2}) + (\frac{1}{2})

extreme f(x)=(3x^2)/(x^2+3)

e x t re m e f (x) = \frac{3 x ^{2}}{x ^{2} + 3}

extreme (x^3)/3-(3x^2)/2-4x+8

e x t re m e \frac{x ^{3}}{3} - \frac{3 x ^{2}}{2} - 4 x + 8

extreme f(x,y)=x^3+9y^4-20x^2y+2

e x t re m e f (x, y) = x^{3} + 9 y^{4} - 20 x^{2} y + 2

extreme 2x^3+3x^2+4

e x t re m e 2 x^{3} + 3 x^{2} + 4