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受欢迎的函数 >

extreme f(x)=x^{1/3}(x^2-9),-4<= x<= 2

解答

极值点

f (x) = x^{\frac{1}{3}} (x^{2} - 9), - 4 \leq x \leq 2

解答

极小值 (- 4, - 7 \cdot 4^{\frac{1}{3}}), 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}}}), 极大值 (2, - 5 \cdot 2^{\frac{1}{3}})

求解步骤

找到临界点:

x = - 4, x = - \frac{3}{7 ^{\frac{1}{2}}}, x = 0, x = \frac{3}{7 ^{\frac{1}{2}}}, x = 2

x^{\frac{1}{3}} (x^{2} - 9), - 4 \leq x \leq 2

的定义域

: - 4 \leq x \leq 2

Find intervals:

递增

: - 4 < 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 < 2

将极值点

x = - 4

代入

x^{\frac{1}{3}} (x^{2} - 9) \Rightarrow y = - 7 4^{\frac{1}{3}}

极小值 (- 4, - 7 \cdot 4^{\frac{1}{3}})

将极值点

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

代入

x^{\frac{1}{3}} (x^{2} - 9) \Rightarrow y = \frac{54 3 ^{\frac{1}{3}}}{7 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}}})

将极值点

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

代入

x^{\frac{1}{3}} (x^{2} - 9) \Rightarrow y = - \frac{54 3 ^{\frac{1}{3}}}{7 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}}})

将极值点

x = 2

代入

x^{\frac{1}{3}} (x^{2} - 9) \Rightarrow y = - 5 2^{\frac{1}{3}}

极大值 (2, - 5 \cdot 2^{\frac{1}{3}})

极小值 (- 4, - 7 \cdot 4^{\frac{1}{3}}), 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}}}), 极大值 (2, - 5 \cdot 2^{\frac{1}{3}})

作图

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