What is the extreme 2x^3+3x^2-12x+5,0<= x<= 2 ?

The extreme 2x^3+3x^2-12x+5,0<= x<= 2 is Maximum(0,5),Minimum(1,-2),Maximum(2,9)

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extreme 2x^3+3x^2-12x+5,0<= x<= 2

extreme points

2 x^{3} + 3 x^{2} - 12 x + 5, 0 \leq x \leq 2

Maximum (0, 5), Minimum (1, - 2), Maximum (2, 9)

Solution steps

Find the critical points:

x = 0, x = 1, x = 2

Domain of

2 x^{3} + 3 x^{2} - 12 x + 5, 0 \leq x \leq 2 : 0 \leq x \leq 2

Find intervals:

Decreasing

: 0 < x < 1,

Increasing

: 1 < x < 2

Plug the extreme point

x = 0

into

2 x^{3} + 3 x^{2} - 12 x + 5 \Rightarrow y = 5

Maximum (0, 5)

Plug the extreme point

x = 1

into

2 x^{3} + 3 x^{2} - 12 x + 5 \Rightarrow y = - 2

Minimum (1, - 2)

Plug the extreme point

x = 2

into

2 x^{3} + 3 x^{2} - 12 x + 5 \Rightarrow y = 9

Maximum (2, 9)

Maximum (0, 5), Minimum (1, - 2), Maximum (2, 9)

f (x, y) = x^{2} + x y + y^{2} - 28 y + 261

(x - 2) (x - 3)^{2}

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

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

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

What is the extreme 2x^3+3x^2-12x+5,0<= x<= 2 ?
The extreme 2x^3+3x^2-12x+5,0<= x<= 2 is Maximum(0,5),Minimum(1,-2),Maximum(2,9)