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Popular >

extreme f(x)=x^2(x-2)^2(x-1)^2

Solution

extreme points

f (x) = x^{2} (x - 2)^{2} (x - 1)^{2}

Solution

Minimum (0, 0), Maximum (\frac{3 - 3}{3}, \frac{4}{27}), Minimum (1, 0), Maximum (\frac{3 + 3}{3}, \frac{4}{27}), Minimum (2, 0)

Solution steps

f^{'} (x) = 2 x (x - 2) (x - 1) (3 x^{2} - 6 x + 2)

Find intervals:

Decreasing

: - \infty < x < 0,

Increasing

: 0 < x < \frac{3 - 1}{3},

Decreasing

: \frac{3 - 1}{3} < x < 1,

Increasing

: 1 < x < \frac{1 + 3}{3},

Decreasing

: \frac{1 + 3}{3} < x < 2,

Increasing

: 2 < x < \infty

Plug

x = 0

into

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

Minimum (0, 0)

Plug

x = \frac{3 - 3}{3}

into

x^{2} (x - 2)^{2} (x - 1)^{2} : \frac{4}{27}

Maximum (\frac{3 - 3}{3}, \frac{4}{27})

Plug

x = 1

into

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

Minimum (1, 0)

Plug

x = \frac{3 + 3}{3}

into

x^{2} (x - 2)^{2} (x - 1)^{2} : \frac{4}{27}

Maximum (\frac{3 + 3}{3}, \frac{4}{27})

Plug

x = 2

into

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

Minimum (2, 0)

Minimum (0, 0), Maximum (\frac{3 - 3}{3}, \frac{4}{27}), Minimum (1, 0), Maximum (\frac{3 + 3}{3}, \frac{4}{27}), Minimum (2, 0)

Popular Examples

asymptotes of f(x)=((x^2+1))/(x^3+1)asymptotes

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

inverse of f(x)= 1/27 x^3 inverse

f (x) = \frac{1}{27} x^{3}

domain of f(x)=sqrt(x^2-2x-8)domain

f (x) = x^{2} - 2 x - 8

shift 4/5 sin(-1/3 x)shift

\frac{4}{5} sin (- \frac{1}{3} x)

domain of f(x)=sqrt(x^4-256)domain

f (x) = x^{4} - 256

Frequently Asked Questions (FAQ)

What is the extreme f(x)=x^2(x-2)^2(x-1)^2 ?
The extreme f(x)=x^2(x-2)^2(x-1)^2 is