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受欢迎的函数 >

extreme f(x)=-x^{2/3}(x-3),-3<= x<= 3

解答

极值点

f (x) = - x^{\frac{2}{3}} (x - 3), - 3 \leq x \leq 3

解答

极大值 (- 3, 6 (- 3)^{\frac{2}{3}}), 极小值 (0, 0), 极大值 (\frac{6}{5}, \frac{9 \cdot 6 ^{\frac{2}{3}}}{5 \cdot 5 ^{\frac{2}{3}}}), 极小值 (3, 0)

求解步骤

找到临界点:

x = - 3, x = 0, x = \frac{6}{5}, x = 3

- x^{\frac{2}{3}} (x - 3), - 3 \leq x \leq 3

的定义域

: - 3 \leq x \leq 3

Find intervals:

递减

: - 3 < x < 0,

递增

: 0 < x < \frac{6}{5},

递减

: \frac{6}{5} < x < 3

将极值点

x = - 3

代入

- x^{\frac{2}{3}} (x - 3) \Rightarrow y = 6 (- 3)^{\frac{2}{3}}

极大值 (- 3, 6 (- 3)^{\frac{2}{3}})

将极值点

x = 0

代入

- x^{\frac{2}{3}} (x - 3) \Rightarrow y = 0

极小值 (0, 0)

将极值点

x = \frac{6}{5}

代入

- x^{\frac{2}{3}} (x - 3) \Rightarrow y = \frac{9 6 ^{\frac{2}{3}}}{5 5 ^{\frac{2}{3}}}

极大值 (\frac{6}{5}, \frac{9 \cdot 6 ^{\frac{2}{3}}}{5 \cdot 5 ^{\frac{2}{3}}})

将极值点

x = 3

代入

- x^{\frac{2}{3}} (x - 3) \Rightarrow y = 0

极小值 (3, 0)

极大值 (- 3, 6 (- 3)^{\frac{2}{3}}), 极小值 (0, 0), 极大值 (\frac{6}{5}, \frac{9 \cdot 6 ^{\frac{2}{3}}}{5 \cdot 5 ^{\frac{2}{3}}}), 极小值 (3, 0)

作图

流行的例子

extreme x^{3/5}

e x t re m e x^{\frac{3}{5}}

extreme f(x)=(8.7x+0.2)/(x^2-4.5x+7.5)

e x t re m e f (x) = \frac{8.7 x + 0.2}{x ^{2} - 4.5 x + 7.5}

extreme f(t)=(e^t)/(t^2+1),(0,3)

e x t re m e f (t) = \frac{e ^{t}}{t ^{2} + 1}, (0, 3)

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e x t re m e f (x) = 5 x^{3} - 5 x^{2} - 5 x + 7

extreme x^2-2x+1

e x t re m e x^{2} - 2 x + 1