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Popular >

extreme f(x)=x^{1/5}(x^2-9)

Solution

extreme points

f (x) = x^{\frac{1}{5}} (x^{2} - 9)

Solution

Maximum (- \frac{3}{1 1 ^{\frac{1}{2}}}, \frac{90 \cdot 3 ^{\frac{1}{5}}}{11 \cdot 1 1 ^{\frac{1}{10}}}), Minimum (\frac{3}{1 1 ^{\frac{1}{2}}}, - \frac{90 \cdot 3 ^{\frac{1}{5}}}{11 \cdot 1 1 ^{\frac{1}{10}}})

Solution steps

f^{'} (x) = \frac{( 1 1 ^{\frac{1}{2}} x + 3 ) ( 1 1 ^{\frac{1}{2}} x - 3 )}{5 x ^{\frac{4}{5}}}

Find intervals:

Increasing

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

Decreasing

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

Decreasing

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

Increasing

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

Plug

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

into

x^{\frac{1}{5}} (x^{2} - 9) : \frac{90 \cdot 3 ^{\frac{1}{5}}}{11 \cdot 1 1 ^{\frac{1}{10}}}

Maximum (- \frac{3}{1 1 ^{\frac{1}{2}}}, \frac{90 \cdot 3 ^{\frac{1}{5}}}{11 \cdot 1 1 ^{\frac{1}{10}}})

Plug

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

into

x^{\frac{1}{5}} (x^{2} - 9) : - \frac{90 \cdot 3 ^{\frac{1}{5}}}{11 \cdot 1 1 ^{\frac{1}{10}}}

Minimum (\frac{3}{1 1 ^{\frac{1}{2}}}, - \frac{90 \cdot 3 ^{\frac{1}{5}}}{11 \cdot 1 1 ^{\frac{1}{10}}})

Maximum (- \frac{3}{1 1 ^{\frac{1}{2}}}, \frac{90 \cdot 3 ^{\frac{1}{5}}}{11 \cdot 1 1 ^{\frac{1}{10}}}), Minimum (\frac{3}{1 1 ^{\frac{1}{2}}}, - \frac{90 \cdot 3 ^{\frac{1}{5}}}{11 \cdot 1 1 ^{\frac{1}{10}}})

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Frequently Asked Questions (FAQ)

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