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Popular Functions >

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

Solution

extreme points

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

Solution

Minimum - \frac{3}{2}, - \frac{27 ( - \frac{3}{2} ) ^{\frac{2}{3}}}{4}, Maximum (0, 0), Minimum (\frac{3}{2}, - \frac{27 \cdot 3 ^{\frac{2}{3}}}{4 \cdot 2 ^{\frac{2}{3}}})

Solution steps

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

Find intervals:

Decreasing

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

Increasing

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

Decreasing

: 0 < x < \frac{3}{2},

Increasing

: \frac{3}{2} < x < \infty

Plug

x = - \frac{3}{2}

into

x^{\frac{2}{3}} (x^{2} - 9) : - \frac{27 ( - \frac{3}{2} ) ^{\frac{2}{3}}}{4}

Minimum - \frac{3}{2}, - \frac{27 ( - \frac{3}{2} ) ^{\frac{2}{3}}}{4}

Plug

x = 0

into

x^{\frac{2}{3}} (x^{2} - 9) : 0

Maximum (0, 0)

Plug

x = \frac{3}{2}

into

x^{\frac{2}{3}} (x^{2} - 9) : - \frac{27 \cdot 3 ^{\frac{2}{3}}}{4 \cdot 2 ^{\frac{2}{3}}}

Minimum (\frac{3}{2}, - \frac{27 \cdot 3 ^{\frac{2}{3}}}{4 \cdot 2 ^{\frac{2}{3}}})

Minimum - \frac{3}{2}, - \frac{27 ( - \frac{3}{2} ) ^{\frac{2}{3}}}{4}, Maximum (0, 0), Minimum (\frac{3}{2}, - \frac{27 \cdot 3 ^{\frac{2}{3}}}{4 \cdot 2 ^{\frac{2}{3}}})

Graph

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Frequently Asked Questions (FAQ)

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