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

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

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

extreme

Minimum (- 4, - 7 \cdot 4^{\frac{1}{3}}), Maximum (- \frac{3}{7 ^{\frac{1}{2}}}, \frac{54 \cdot 3 ^{\frac{1}{3}}}{7 \cdot 7 ^{\frac{1}{6}}}), Minimum (\frac{3}{7 ^{\frac{1}{2}}}, - \frac{54 \cdot 3 ^{\frac{1}{3}}}{7 \cdot 7 ^{\frac{1}{6}}}), Maximum (2, - 5 \cdot 2^{\frac{1}{3}})

Solution steps

Find the critical points:

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

Domain of

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

Find intervals:

Increasing

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

Decreasing

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

Decreasing

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

Increasing

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

Plug the extreme point

x = - 4

into

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

Minimum (- 4, - 7 \cdot 4^{\frac{1}{3}})

Plug the extreme point

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

into

x^{\frac{1}{3}} (x^{2} - 9) \Rightarrow y = \frac{54 3 ^{\frac{1}{3}}}{7 7 ^{\frac{1}{6}}}

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

Plug the extreme point

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

into

x^{\frac{1}{3}} (x^{2} - 9) \Rightarrow y = - \frac{54 3 ^{\frac{1}{3}}}{7 7 ^{\frac{1}{6}}}

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

Plug the extreme point

x = 2

into

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

Maximum (2, - 5 \cdot 2^{\frac{1}{3}})

Minimum (- 4, - 7 \cdot 4^{\frac{1}{3}}), Maximum (- \frac{3}{7 ^{\frac{1}{2}}}, \frac{54 \cdot 3 ^{\frac{1}{3}}}{7 \cdot 7 ^{\frac{1}{6}}}), Minimum (\frac{3}{7 ^{\frac{1}{2}}}, - \frac{54 \cdot 3 ^{\frac{1}{3}}}{7 \cdot 7 ^{\frac{1}{6}}}), Maximum (2, - 5 \cdot 2^{\frac{1}{3}})

What is the extreme f(x)=x^{1/3}(x^2-9),-4<= x<= 2 ?
The extreme f(x)=x^{1/3}(x^2-9),-4<= x<= 2 is