What is the extreme f(x)=x^3-6x^2+5,-2<= x<= 6 ?

The extreme f(x)=x^3-6x^2+5,-2<= x<= 6 is Minimum(-2,-27),Maximum(0,5),Minimum(4,-27),Maximum(6,5)

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extreme f(x)=x^3-6x^2+5,-2<= x<= 6

extreme points

f (x) = x^{3} - 6 x^{2} + 5, - 2 \leq x \leq 6

Minimum (- 2, - 27), Maximum (0, 5), Minimum (4, - 27), Maximum (6, 5)

Solution steps

Find the critical points:

x = - 2, x = 0, x = 4, x = 6

Domain of

x^{3} - 6 x^{2} + 5, - 2 \leq x \leq 6 : - 2 \leq x \leq 6

Find intervals:

Increasing

: - 2 < x < 0,

Decreasing

: 0 < x < 4,

Increasing

: 4 < x < 6

Plug the extreme point

x = - 2

into

x^{3} - 6 x^{2} + 5 \Rightarrow y = - 27

Minimum (- 2, - 27)

Plug the extreme point

x = 0

into

x^{3} - 6 x^{2} + 5 \Rightarrow y = 5

Maximum (0, 5)

Plug the extreme point

x = 4

into

x^{3} - 6 x^{2} + 5 \Rightarrow y = - 27

Minimum (4, - 27)

Plug the extreme point

x = 6

into

x^{3} - 6 x^{2} + 5 \Rightarrow y = 5

Maximum (6, 5)

Minimum (- 2, - 27), Maximum (0, 5), Minimum (4, - 27), Maximum (6, 5)

f (x) = (41000 - 5000 x) (2.5 + 0.5 x)

f (x) = - 12 x + 30

x^{3} + 15 x^{2}

f (x) = 60 x^{4} - 60 x^{3} - 60 x^{2}

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

What is the extreme f(x)=x^3-6x^2+5,-2<= x<= 6 ?
The extreme f(x)=x^3-6x^2+5,-2<= x<= 6 is Minimum(-2,-27),Maximum(0,5),Minimum(4,-27),Maximum(6,5)