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Popular >

extreme f(x)=(x-1)(x-2)^2(x-3)^5

Solution

extreme points

f (x) = (x - 1) (x - 2)^{2} (x - 3)^{5}

Solution

Minimum (\frac{7 - 5}{4}, - \frac{243}{8} + \frac{11 ( 1357 - 25 5 )}{512}), Maximum (2, 0), Minimum (\frac{7 + 5}{4}, - \frac{243}{8} + \frac{11 ( 1357 + 25 5 )}{512}), Saddle (3, 0)

Solution steps

f^{'} (x) = 2 (x - 3)^{4} (x - 2) (4 x^{2} - 14 x + 11)

Find intervals:

Decreasing

: - \infty < x < \frac{7 - 5}{4},

Increasing

: \frac{7 - 5}{4} < x < 2,

Decreasing

: 2 < x < \frac{7 + 5}{4},

Increasing

: \frac{7 + 5}{4} < x < 3,

Increasing

: 3 < x < \infty

Plug

x = \frac{7 - 5}{4}

into

(x - 1) (x - 2)^{2} (x - 3)^{5} : - \frac{243}{8} + \frac{11 ( 1357 - 25 5 )}{512}

Minimum (\frac{7 - 5}{4}, - \frac{243}{8} + \frac{11 ( 1357 - 25 5 )}{512})

Plug

x = 2

into

(x - 1) (x - 2)^{2} (x - 3)^{5} : 0

Maximum (2, 0)

Plug

x = \frac{7 + 5}{4}

into

(x - 1) (x - 2)^{2} (x - 3)^{5} : - \frac{243}{8} + \frac{11 ( 1357 + 25 5 )}{512}

Minimum (\frac{7 + 5}{4}, - \frac{243}{8} + \frac{11 ( 1357 + 25 5 )}{512})

Plug

x = 3

into

(x - 1) (x - 2)^{2} (x - 3)^{5} : 0

Saddle (3, 0)

Minimum (\frac{7 - 5}{4}, - \frac{243}{8} + \frac{11 ( 1357 - 25 5 )}{512}), Maximum (2, 0), Minimum (\frac{7 + 5}{4}, - \frac{243}{8} + \frac{11 ( 1357 + 25 5 )}{512}), Saddle (3, 0)

Popular Examples

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f (x, y) = 5 x^{2} + 5 y^{2} - 5 x y - 10 x - 5 y + 18

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f (x) = 355 x^{2} - 1420 x^{3}

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extreme f(x)=5x^4-x^5+5 extreme points

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

Frequently Asked Questions (FAQ)

What is the extreme f(x)=(x-1)(x-2)^2(x-3)^5 ?
The extreme f(x)=(x-1)(x-2)^2(x-3)^5 is