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Popular Functions >

extreme f(x)=-2x^3+27x^2-108x,2<= x<= 7

Solution

extreme points

f (x) = - 2 x^{3} + 27 x^{2} - 108 x, 2 \leq x \leq 7

Solution

Maximum (2, - 124), Minimum (3, - 135), Maximum (6, - 108), Minimum (7, - 119)

Solution steps

Find the critical points:

x = 2, x = 3, x = 6, x = 7

Domain of

- 2 x^{3} + 27 x^{2} - 108 x, 2 \leq x \leq 7 : 2 \leq x \leq 7

Find intervals:

Decreasing

: 2 < x < 3,

Increasing

: 3 < x < 6,

Decreasing

: 6 < x < 7

Plug the extreme point

x = 2

into

- 2 x^{3} + 27 x^{2} - 108 x \Rightarrow y = - 124

Maximum (2, - 124)

Plug the extreme point

x = 3

into

- 2 x^{3} + 27 x^{2} - 108 x \Rightarrow y = - 135

Minimum (3, - 135)

Plug the extreme point

x = 6

into

- 2 x^{3} + 27 x^{2} - 108 x \Rightarrow y = - 108

Maximum (6, - 108)

Plug the extreme point

x = 7

into

- 2 x^{3} + 27 x^{2} - 108 x \Rightarrow y = - 119

Minimum (7, - 119)

Maximum (2, - 124), Minimum (3, - 135), Maximum (6, - 108), Minimum (7, - 119)

Graph

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