What is the extreme f(x)=2x^3-3x^2+1,-1<= x<= 3 ?

The extreme f(x)=2x^3-3x^2+1,-1<= x<= 3 is Minimum(-1,-4),Maximum(0,1),Minimum(1,0),Maximum(3,28)

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extreme f(x)=2x^3-3x^2+1,-1<= x<= 3

extreme points

f (x) = 2 x^{3} - 3 x^{2} + 1, - 1 \leq x \leq 3

Minimum (- 1, - 4), Maximum (0, 1), Minimum (1, 0), Maximum (3, 28)

Solution steps

Find the critical points:

x = - 1, x = 0, x = 1, x = 3

Domain of

2 x^{3} - 3 x^{2} + 1, - 1 \leq x \leq 3 : - 1 \leq x \leq 3

Find intervals:

Increasing

: - 1 < x < 0,

Decreasing

: 0 < x < 1,

Increasing

: 1 < x < 3

Plug the extreme point

x = - 1

into

2 x^{3} - 3 x^{2} + 1 \Rightarrow y = - 4

Minimum (- 1, - 4)

Plug the extreme point

x = 0

into

2 x^{3} - 3 x^{2} + 1 \Rightarrow y = 1

Maximum (0, 1)

Plug the extreme point

x = 1

into

2 x^{3} - 3 x^{2} + 1 \Rightarrow y = 0

Minimum (1, 0)

Plug the extreme point

x = 3

into

2 x^{3} - 3 x^{2} + 1 \Rightarrow y = 28

Maximum (3, 28)

Minimum (- 1, - 4), Maximum (0, 1), Minimum (1, 0), Maximum (3, 28)

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

f (x) = x^{4} - 10 x^{2} + 9

f (x) = x^{2} y + y^{3} - 48 y

x^{2} y - 2 x y + 3 y^{3} - 3 y

\frac{1}{1 + x ^{2}}

What is the extreme f(x)=2x^3-3x^2+1,-1<= x<= 3 ?
The extreme f(x)=2x^3-3x^2+1,-1<= x<= 3 is Minimum(-1,-4),Maximum(0,1),Minimum(1,0),Maximum(3,28)