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Popular >

extreme f(x)=x^5-4x^3+4x-1

Solution

extreme points

f (x) = x^{5} - 4 x^{3} + 4 x - 1

Solution

Maximum (- 2, - 1), Minimum (- \frac{2}{5}, - (\frac{2}{5})^{\frac{5}{2}} + 4 (\frac{2}{5})^{\frac{3}{2}} - 4 \frac{2}{5} - 1), Maximum (\frac{2}{5}, (\frac{2}{5})^{\frac{5}{2}} - 4 (\frac{2}{5})^{\frac{3}{2}} + 4 \frac{2}{5} - 1), Minimum (2, - 1)

Solution steps

f^{'} (x) = 5 x^{4} - 12 x^{2} + 4

Find intervals:

Increasing

: - \infty < x < - 2,

Decreasing

: - 2 < x < - \frac{2}{5},

Increasing

: - \frac{2}{5} < x < \frac{2}{5},

Decreasing

: \frac{2}{5} < x < 2,

Increasing

: 2 < x < \infty

Plug

x = - 2

into

x^{5} - 4 x^{3} + 4 x - 1 : - 1

Maximum (- 2, - 1)

Plug

x = - \frac{2}{5}

into

x^{5} - 4 x^{3} + 4 x - 1 : - (\frac{2}{5})^{\frac{5}{2}} + 4 (\frac{2}{5})^{\frac{3}{2}} - 4 \frac{2}{5} - 1

Minimum (- \frac{2}{5}, - (\frac{2}{5})^{\frac{5}{2}} + 4 (\frac{2}{5})^{\frac{3}{2}} - 4 \frac{2}{5} - 1)

Plug

x = \frac{2}{5}

into

x^{5} - 4 x^{3} + 4 x - 1 : (\frac{2}{5})^{\frac{5}{2}} - 4 (\frac{2}{5})^{\frac{3}{2}} + 4 \frac{2}{5} - 1

Maximum (\frac{2}{5}, (\frac{2}{5})^{\frac{5}{2}} - 4 (\frac{2}{5})^{\frac{3}{2}} + 4 \frac{2}{5} - 1)

Plug

x = 2

into

x^{5} - 4 x^{3} + 4 x - 1 : - 1

Minimum (2, - 1)

Maximum (- 2, - 1), Minimum (- \frac{2}{5}, - (\frac{2}{5})^{\frac{5}{2}} + 4 (\frac{2}{5})^{\frac{3}{2}} - 4 \frac{2}{5} - 1), Maximum (\frac{2}{5}, (\frac{2}{5})^{\frac{5}{2}} - 4 (\frac{2}{5})^{\frac{3}{2}} + 4 \frac{2}{5} - 1), Minimum (2, - 1)

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Frequently Asked Questions (FAQ)

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