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Popular >

extreme f(x)=3x^5-6x^3

Solution

extreme points

f (x) = 3 x^{5} - 6 x^{3}

Solution

Maximum (- \frac{6}{5}, - 3 (\frac{6}{5})^{\frac{5}{2}} + 6 (\frac{6}{5})^{\frac{3}{2}}), Saddle (0, 0), Minimum (\frac{6}{5}, 3 (\frac{6}{5})^{\frac{5}{2}} - 6 (\frac{6}{5})^{\frac{3}{2}})

Solution steps

f^{'} (x) = 15 x^{4} - 18 x^{2}

Find intervals:

Increasing

: - \infty < x < - \frac{6}{5},

Decreasing

: - \frac{6}{5} < x < 0,

Decreasing

: 0 < x < \frac{6}{5},

Increasing

: \frac{6}{5} < x < \infty

Plug

x = - \frac{6}{5}

into

3 x^{5} - 6 x^{3} : - 3 (\frac{6}{5})^{\frac{5}{2}} + 6 (\frac{6}{5})^{\frac{3}{2}}

Maximum (- \frac{6}{5}, - 3 (\frac{6}{5})^{\frac{5}{2}} + 6 (\frac{6}{5})^{\frac{3}{2}})

Plug

x = 0

into

3 x^{5} - 6 x^{3} : 0

Saddle (0, 0)

Plug

x = \frac{6}{5}

into

3 x^{5} - 6 x^{3} : 3 (\frac{6}{5})^{\frac{5}{2}} - 6 (\frac{6}{5})^{\frac{3}{2}}

Minimum (\frac{6}{5}, 3 (\frac{6}{5})^{\frac{5}{2}} - 6 (\frac{6}{5})^{\frac{3}{2}})

Maximum (- \frac{6}{5}, - 3 (\frac{6}{5})^{\frac{5}{2}} + 6 (\frac{6}{5})^{\frac{3}{2}}), Saddle (0, 0), Minimum (\frac{6}{5}, 3 (\frac{6}{5})^{\frac{5}{2}} - 6 (\frac{6}{5})^{\frac{3}{2}})

Popular Examples

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Frequently Asked Questions (FAQ)

What is the extreme f(x)=3x^5-6x^3 ?
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