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受欢迎的函数 >

extreme f(x)=x^{1/3}(x^2-9)

解答

极值点

f (x) = x^{\frac{1}{3}} (x^{2} - 9)

解答

极大值 (- \frac{3}{7 ^{\frac{1}{2}}}, \frac{54 \cdot 3 ^{\frac{1}{3}}}{7 \cdot 7 ^{\frac{1}{6}}}), 极小值 (\frac{3}{7 ^{\frac{1}{2}}}, - \frac{54 \cdot 3 ^{\frac{1}{3}}}{7 \cdot 7 ^{\frac{1}{6}}})

求解步骤

f^{'} (x) = \frac{( 7 ^{\frac{1}{2}} x + 3 ) ( 7 ^{\frac{1}{2}} x - 3 )}{3 x ^{\frac{2}{3}}}

Find intervals:

递增

: - \infty < x < - \frac{3}{7 ^{\frac{1}{2}}},

递减

: - \frac{3}{7 ^{\frac{1}{2}}} < x < 0,

递减

: 0 < x < \frac{3}{7 ^{\frac{1}{2}}},

递增

: \frac{3}{7 ^{\frac{1}{2}}} < x < \infty

将

x = - \frac{3}{7 ^{\frac{1}{2}}}

代入

x^{\frac{1}{3}} (x^{2} - 9) : \frac{54 \cdot 3 ^{\frac{1}{3}}}{7 \cdot 7 ^{\frac{1}{6}}}

极大值 (- \frac{3}{7 ^{\frac{1}{2}}}, \frac{54 \cdot 3 ^{\frac{1}{3}}}{7 \cdot 7 ^{\frac{1}{6}}})

将

x = \frac{3}{7 ^{\frac{1}{2}}}

代入

x^{\frac{1}{3}} (x^{2} - 9) : - \frac{54 \cdot 3 ^{\frac{1}{3}}}{7 \cdot 7 ^{\frac{1}{6}}}

极小值 (\frac{3}{7 ^{\frac{1}{2}}}, - \frac{54 \cdot 3 ^{\frac{1}{3}}}{7 \cdot 7 ^{\frac{1}{6}}})

极大值 (- \frac{3}{7 ^{\frac{1}{2}}}, \frac{54 \cdot 3 ^{\frac{1}{3}}}{7 \cdot 7 ^{\frac{1}{6}}}), 极小值 (\frac{3}{7 ^{\frac{1}{2}}}, - \frac{54 \cdot 3 ^{\frac{1}{3}}}{7 \cdot 7 ^{\frac{1}{6}}})

作图

流行的例子

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e x t re m e f (x, y) = 3 x^{2} + 2 y^{2} - x y - 4 x - 7 y + 12

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

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