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Popular >

extreme f(x)=-2(x-2)^3(x+2)(x+4)

Solution

extreme points

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

Solution

Minimum (- \frac{2 ( 5 + 10 )}{5}, - \frac{2 ( 4352 2 + 2240 5 )}{25 5} + 128), Maximum (\frac{2 ( 10 - 5 )}{5}, - \frac{2 ( 2240 5 - 4352 2 )}{25 5} + 128), Saddle (2, 0)

Solution steps

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

Find intervals:

Decreasing

: - \infty < x < - \frac{2 ( 5 + 10 )}{5},

Increasing

: - \frac{2 ( 5 + 10 )}{5} < x < \frac{2 ( 10 - 5 )}{5},

Decreasing

: \frac{2 ( 10 - 5 )}{5} < x < 2,

Decreasing

: 2 < x < \infty

Plug

x = - \frac{2 ( 5 + 10 )}{5}

into

- 2 (x - 2)^{3} (x + 2) (x + 4) : - \frac{2 ( 4352 2 + 2240 5 )}{25 5} + 128

Minimum (- \frac{2 ( 5 + 10 )}{5}, - \frac{2 ( 4352 2 + 2240 5 )}{25 5} + 128)

Plug

x = \frac{2 ( 10 - 5 )}{5}

into

- 2 (x - 2)^{3} (x + 2) (x + 4) : - \frac{2 ( 2240 5 - 4352 2 )}{25 5} + 128

Maximum (\frac{2 ( 10 - 5 )}{5}, - \frac{2 ( 2240 5 - 4352 2 )}{25 5} + 128)

Plug

x = 2

into

- 2 (x - 2)^{3} (x + 2) (x + 4) : 0

Saddle (2, 0)

Minimum (- \frac{2 ( 5 + 10 )}{5}, - \frac{2 ( 4352 2 + 2240 5 )}{25 5} + 128), Maximum (\frac{2 ( 10 - 5 )}{5}, - \frac{2 ( 2240 5 - 4352 2 )}{25 5} + 128), Saddle (2, 0)

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Frequently Asked Questions (FAQ)

What is the extreme f(x)=-2(x-2)^3(x+2)(x+4) ?
The extreme f(x)=-2(x-2)^3(x+2)(x+4) is