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Popular >

extreme f(x)=x^{4/3}-x^{2/3}

Solution

extreme points

f (x) = x^{\frac{4}{3}} - x^{\frac{2}{3}}

Solution

Minimum (- \frac{1}{2 \cdot 2 ^{\frac{1}{2}}}, (- \frac{1}{2 \cdot 2 ^{\frac{1}{2}}})^{\frac{4}{3}} - (- \frac{1}{2 \cdot 2 ^{\frac{1}{2}}})^{\frac{2}{3}}), Maximum (0, 0), Minimum (\frac{1}{2 \cdot 2 ^{\frac{1}{2}}}, (\frac{1}{2 \cdot 2 ^{\frac{1}{2}}})^{\frac{4}{3}} - (\frac{1}{2 \cdot 2 ^{\frac{1}{2}}})^{\frac{2}{3}})

Solution steps

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

Find intervals:

Decreasing

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

Increasing

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

Decreasing

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

Increasing

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

Plug

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

into

x^{\frac{4}{3}} - x^{\frac{2}{3}} : (- \frac{1}{2 \cdot 2 ^{\frac{1}{2}}})^{\frac{4}{3}} - (- \frac{1}{2 \cdot 2 ^{\frac{1}{2}}})^{\frac{2}{3}}

Minimum (- \frac{1}{2 \cdot 2 ^{\frac{1}{2}}}, (- \frac{1}{2 \cdot 2 ^{\frac{1}{2}}})^{\frac{4}{3}} - (- \frac{1}{2 \cdot 2 ^{\frac{1}{2}}})^{\frac{2}{3}})

Plug

x = 0

into

x^{\frac{4}{3}} - x^{\frac{2}{3}} : 0

Maximum (0, 0)

Plug

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

into

x^{\frac{4}{3}} - x^{\frac{2}{3}} : (\frac{1}{2 \cdot 2 ^{\frac{1}{2}}})^{\frac{4}{3}} - (\frac{1}{2 \cdot 2 ^{\frac{1}{2}}})^{\frac{2}{3}}

Minimum (\frac{1}{2 \cdot 2 ^{\frac{1}{2}}}, (\frac{1}{2 \cdot 2 ^{\frac{1}{2}}})^{\frac{4}{3}} - (\frac{1}{2 \cdot 2 ^{\frac{1}{2}}})^{\frac{2}{3}})

Minimum (- \frac{1}{2 \cdot 2 ^{\frac{1}{2}}}, (- \frac{1}{2 \cdot 2 ^{\frac{1}{2}}})^{\frac{4}{3}} - (- \frac{1}{2 \cdot 2 ^{\frac{1}{2}}})^{\frac{2}{3}}), Maximum (0, 0), Minimum (\frac{1}{2 \cdot 2 ^{\frac{1}{2}}}, (\frac{1}{2 \cdot 2 ^{\frac{1}{2}}})^{\frac{4}{3}} - (\frac{1}{2 \cdot 2 ^{\frac{1}{2}}})^{\frac{2}{3}})

Graph

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Frequently Asked Questions (FAQ)

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