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Popular >

extreme f(x)=x(x+2)^2(x-3)^3

Solution

extreme points

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

Solution

Minimum (- 2, 0), Maximum (\frac{1 - 145}{12}, \frac{25 ( 21257 145 - 14705 )}{93312}), Minimum (\frac{1 + 145}{12}, - \frac{25 ( 21257 145 + 14705 )}{93312}), Saddle (3, 0)

Solution steps

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

Find intervals:

Decreasing

: - \infty < x < - 2,

Increasing

: - 2 < x < \frac{1 - 145}{12},

Decreasing

: \frac{1 - 145}{12} < x < \frac{1 + 145}{12},

Increasing

: \frac{1 + 145}{12} < x < 3,

Increasing

: 3 < x < \infty

Plug

x = - 2

into

x (x + 2)^{2} (x - 3)^{3} : 0

Minimum (- 2, 0)

Plug

x = \frac{1 - 145}{12}

into

x (x + 2)^{2} (x - 3)^{3} : \frac{25 ( 21257 145 - 14705 )}{93312}

Maximum (\frac{1 - 145}{12}, \frac{25 ( 21257 145 - 14705 )}{93312})

Plug

x = \frac{1 + 145}{12}

into

x (x + 2)^{2} (x - 3)^{3} : - \frac{25 ( 21257 145 + 14705 )}{93312}

Minimum (\frac{1 + 145}{12}, - \frac{25 ( 21257 145 + 14705 )}{93312})

Plug

x = 3

into

x (x + 2)^{2} (x - 3)^{3} : 0

Saddle (3, 0)

Minimum (- 2, 0), Maximum (\frac{1 - 145}{12}, \frac{25 ( 21257 145 - 14705 )}{93312}), Minimum (\frac{1 + 145}{12}, - \frac{25 ( 21257 145 + 14705 )}{93312}), Saddle (3, 0)

Graph

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Frequently Asked Questions (FAQ)

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