What is the extreme x^4-4/3 x^3-12x^2 ?

The extreme x^4-4/3 x^3-12x^2 is Minimum(-2,-64/3),Maximum(0,0),Minimum(3,-63)

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extreme x^4-4/3 x^3-12x^2

extreme points

x^{4} - \frac{4}{3} x^{3} - 12 x^{2}

Minimum (- 2, - \frac{64}{3}), Maximum (0, 0), Minimum (3, - 63)

Solution steps

f^{'} (x) = 4 x^{3} - 4 x^{2} - 24 x

Find intervals:

Decreasing

: - \infty < x < - 2,

Increasing

: - 2 < x < 0,

Decreasing

: 0 < x < 3,

Increasing

: 3 < x < \infty

Plug

x = - 2

into

x^{4} - \frac{4}{3} x^{3} - 12 x^{2} : - \frac{64}{3}

Minimum (- 2, - \frac{64}{3})

Plug

x = 0

into

x^{4} - \frac{4}{3} x^{3} - 12 x^{2} : 0

Maximum (0, 0)

Plug

x = 3

into

x^{4} - \frac{4}{3} x^{3} - 12 x^{2} : - 63

Minimum (3, - 63)

Minimum (- 2, - \frac{64}{3}), Maximum (0, 0), Minimum (3, - 63)

x^{2} - 4 x + 12

f (x, y) = 7 + 4 x^{2} + 2 y^{2}

f (x) = - 5 + x - x^{2}

15 y^{30}

f (x) = 3 x^{2} - 30 x + 12, - 10 \leq x \leq 10

What is the extreme x^4-4/3 x^3-12x^2 ?
The extreme x^4-4/3 x^3-12x^2 is Minimum(-2,-64/3),Maximum(0,0),Minimum(3,-63)