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受欢迎的函数 >

extreme f(x)=(x^4)/4-(x/3)^3-x^2+1

解答

极值点

f (x) = \frac{x ^{4}}{4} - (\frac{x}{3})^{3} - x^{2} + 1

解答

极小值 (\frac{1 - 649}{18}, \frac{649 649 - 158437}{157464} + 1), 极大值 (0, 1), 极小值 (\frac{1 + 649}{18}, - \frac{158437 + 649 649}{157464} + 1)

求解步骤

f^{'} (x) = x^{3} - \frac{x ^{2}}{9} - 2 x

Find intervals:

递减

: - \infty < x < \frac{1 - 649}{18},

递增

: \frac{1 - 649}{18} < x < 0,

递减

: 0 < x < \frac{1 + 649}{18},

递增

: \frac{1 + 649}{18} < x < \infty

将

x = \frac{1 - 649}{18}

代入

\frac{x ^{4}}{4} - (\frac{x}{3})^{3} - x^{2} + 1 : \frac{649 649 - 158437}{157464} + 1

极小值 (\frac{1 - 649}{18}, \frac{649 649 - 158437}{157464} + 1)

将

x = 0

代入

\frac{x ^{4}}{4} - (\frac{x}{3})^{3} - x^{2} + 1 : 1

极大值 (0, 1)

将

x = \frac{1 + 649}{18}

代入

\frac{x ^{4}}{4} - (\frac{x}{3})^{3} - x^{2} + 1 : - \frac{158437 + 649 649}{157464} + 1

极小值 (\frac{1 + 649}{18}, - \frac{158437 + 649 649}{157464} + 1)

极小值 (\frac{1 - 649}{18}, \frac{649 649 - 158437}{157464} + 1), 极大值 (0, 1), 极小值 (\frac{1 + 649}{18}, - \frac{158437 + 649 649}{157464} + 1)

作图

流行的例子

extreme f(x)=cos(x)-9x,0<= x<= 4pi

e x t re m e f (x) = cos (x) - 9 x, 0 \leq x \leq 4 π

extreme y=xe^{-(x^2)/(18)}

e x t re m e y = x e^{- \frac{x ^{2}}{18}}

extreme f(x)=(8-9x)/(\sqrt[3]{x+3)}

e x t re m e f (x) = \frac{8 - 9 x}{3 x + 3}

extreme f(x,y)=13-4x+8y

e x t re m e f (x, y) = 13 - 4 x + 8 y

extreme f(x)=-10x^2+60x-80

e x t re m e f (x) = - 10 x^{2} + 60 x - 80