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

The extreme f(x)=x^4(x+2)^2 is Minimum(-2,0),Maximum(-4/3 , 1024/729),Minimum(0,0)

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

extreme points

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

Minimum (- 2, 0), Maximum (- \frac{4}{3}, \frac{1024}{729}), Minimum (0, 0)

Solution steps

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

Find intervals:

Decreasing

: - \infty < x < - 2,

Increasing

: - 2 < x < - \frac{4}{3},

Decreasing

: - \frac{4}{3} < x < 0,

Increasing

: 0 < x < \infty

Plug

x = - 2

into

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

Minimum (- 2, 0)

Plug

x = - \frac{4}{3}

into

x^{4} (x + 2)^{2} : \frac{1024}{729}

Maximum (- \frac{4}{3}, \frac{1024}{729})

Plug

x = 0

into

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

Minimum (0, 0)

Minimum (- 2, 0), Maximum (- \frac{4}{3}, \frac{1024}{729}), Minimum (0, 0)

f (y, x) = \frac{500}{4 + x ^{2} + y ^{2}}

f (x) = \frac{x + 3}{x ^{2} - 4 x - 21}

2.5 x^{2} + 75 x + 25000

f (x) = (100 - 2 x) \cdot (75 - 2 x) \cdot x

What is the extreme f(x)=x^4(x+2)^2 ?
The extreme f(x)=x^4(x+2)^2 is Minimum(-2,0),Maximum(-4/3 , 1024/729),Minimum(0,0)