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Popular >

extreme f(x)=(x^{(2/3)})(1-x^2)

Solution

extreme points

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

Solution

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

Solution steps

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

Find intervals:

Increasing

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

Decreasing

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

Increasing

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

Decreasing

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

Plug

x = - \frac{1}{2}

into

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

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

Plug

x = 0

into

x^{\frac{2}{3}} (1 - x^{2}) : 0

Minimum (0, 0)

Plug

x = \frac{1}{2}

into

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

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

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

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Frequently Asked Questions (FAQ)

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