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Popular >

extreme (1-2x)^4(1-x)^2x^6

Solution

extreme points

(1 - 2 x)^{4} (1 - x)^{2} x^{6}

Solution

Minimum (0, 0), Maximum (\frac{7 - 13}{12}, \frac{312527872 - 85955584 13}{278628139008}), Minimum (\frac{1}{2}, 0), Maximum (\frac{7 + 13}{12}, \frac{312527872 + 85955584 13}{278628139008}), Minimum (1, 0)

Solution steps

f^{'} (x) = - 2 x^{5} (1 - x) (2 x - 1)^{3} (12 x^{2} - 14 x + 3)

Find intervals:

Decreasing

: - \infty < x < 0,

Increasing

: 0 < x < \frac{7 - 13}{12},

Decreasing

: \frac{7 - 13}{12} < x < \frac{1}{2},

Increasing

: \frac{1}{2} < x < \frac{7 + 13}{12},

Decreasing

: \frac{7 + 13}{12} < x < 1,

Increasing

: 1 < x < \infty

Plug

x = 0

into

(1 - 2 x)^{4} (1 - x)^{2} x^{6} : 0

Minimum (0, 0)

Plug

x = \frac{7 - 13}{12}

into

(1 - 2 x)^{4} (1 - x)^{2} x^{6} : \frac{312527872 - 85955584 13}{278628139008}

Maximum (\frac{7 - 13}{12}, \frac{312527872 - 85955584 13}{278628139008})

Plug

x = \frac{1}{2}

into

(1 - 2 x)^{4} (1 - x)^{2} x^{6} : 0

Minimum (\frac{1}{2}, 0)

Plug

x = \frac{7 + 13}{12}

into

(1 - 2 x)^{4} (1 - x)^{2} x^{6} : \frac{312527872 + 85955584 13}{278628139008}

Maximum (\frac{7 + 13}{12}, \frac{312527872 + 85955584 13}{278628139008})

Plug

x = 1

into

(1 - 2 x)^{4} (1 - x)^{2} x^{6} : 0

Minimum (1, 0)

Minimum (0, 0), Maximum (\frac{7 - 13}{12}, \frac{312527872 - 85955584 13}{278628139008}), Minimum (\frac{1}{2}, 0), Maximum (\frac{7 + 13}{12}, \frac{312527872 + 85955584 13}{278628139008}), Minimum (1, 0)

Graph

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Frequently Asked Questions (FAQ)

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