What is the extreme 1/4 x^4-2/3 x^3-1/2 x^2+2x-1 ?

The extreme 1/4 x^4-2/3 x^3-1/2 x^2+2x-1 is Minimum(-1,-31/12),Maximum(1, 1/12),Minimum(2,-1/3)

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extreme 1/4 x^4-2/3 x^3-1/2 x^2+2x-1

extreme points

\frac{1}{4} x^{4} - \frac{2}{3} x^{3} - \frac{1}{2} x^{2} + 2 x - 1

Minimum (- 1, - \frac{31}{12}), Maximum (1, \frac{1}{12}), Minimum (2, - \frac{1}{3})

Solution steps

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

Find intervals:

Decreasing

: - \infty < x < - 1,

Increasing

: - 1 < x < 1,

Decreasing

: 1 < x < 2,

Increasing

: 2 < x < \infty

Plug

x = - 1

into

\frac{1}{4} x^{4} - \frac{2}{3} x^{3} - \frac{1}{2} x^{2} + 2 x - 1 : - \frac{31}{12}

Minimum (- 1, - \frac{31}{12})

Plug

x = 1

into

\frac{1}{4} x^{4} - \frac{2}{3} x^{3} - \frac{1}{2} x^{2} + 2 x - 1 : \frac{1}{12}

Maximum (1, \frac{1}{12})

Plug

x = 2

into

\frac{1}{4} x^{4} - \frac{2}{3} x^{3} - \frac{1}{2} x^{2} + 2 x - 1 : - \frac{1}{3}

Minimum (2, - \frac{1}{3})

Minimum (- 1, - \frac{31}{12}), Maximum (1, \frac{1}{12}), Minimum (2, - \frac{1}{3})

f (x) = 2 x^{2} + 3 y^{2} - 7

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

f (x) = 2 x^{2} + 3 x y + 4 y^{2} - 5 x + 2 y

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

f (x) = - x^{2} + 4 x + 9

What is the extreme 1/4 x^4-2/3 x^3-1/2 x^2+2x-1 ?
The extreme 1/4 x^4-2/3 x^3-1/2 x^2+2x-1 is Minimum(-1,-31/12),Maximum(1, 1/12),Minimum(2,-1/3)