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受欢迎的函数 >

extreme 2/3 x^3-39/2 x^2+19x,[0<= x<= 1]

解答

极值点

\frac{2}{3} x^{3} - \frac{39}{2} x^{2} + 19 x, [0 \leq x \leq 1]

解答

极小值 (0, 0), 极大值 (\frac{1}{2}, \frac{113}{24}), 极小值 (1, \frac{1}{6})

求解步骤

找到临界点:

x = 0, x = \frac{1}{2}, x = 1

\frac{2}{3} x^{3} - \frac{39}{2} x^{2} + 19 x, 0 \leq x \leq 1

的定义域

: 0 \leq x \leq 1

Find intervals:

递增

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

递减

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

将极值点

x = 0

代入

\frac{2}{3} x^{3} - \frac{39}{2} x^{2} + 19 x \Rightarrow y = 0

极小值 (0, 0)

将极值点

x = \frac{1}{2}

代入

\frac{2}{3} x^{3} - \frac{39}{2} x^{2} + 19 x \Rightarrow y = \frac{113}{24}

极大值 (\frac{1}{2}, \frac{113}{24})

将极值点

x = 1

代入

\frac{2}{3} x^{3} - \frac{39}{2} x^{2} + 19 x \Rightarrow y = \frac{1}{6}

极小值 (1, \frac{1}{6})

极小值 (0, 0), 极大值 (\frac{1}{2}, \frac{113}{24}), 极小值 (1, \frac{1}{6})

作图

流行的例子

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