What is the extreme f(x)=2x^3-21x^2+60x,0<= x<= 6 ?

The extreme f(x)=2x^3-21x^2+60x,0<= x<= 6 is Minimum(0,0),Maximum(2,52),Minimum(5,25),Maximum(6,36)

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extreme f(x)=2x^3-21x^2+60x,0<= x<= 6

extreme points

f (x) = 2 x^{3} - 21 x^{2} + 60 x, 0 \leq x \leq 6

Minimum (0, 0), Maximum (2, 52), Minimum (5, 25), Maximum (6, 36)

Solution steps

Find the critical points:

x = 0, x = 2, x = 5, x = 6

Domain of

2 x^{3} - 21 x^{2} + 60 x, 0 \leq x \leq 6 : 0 \leq x \leq 6

Find intervals:

Increasing

: 0 < x < 2,

Decreasing

: 2 < x < 5,

Increasing

: 5 < x < 6

Plug the extreme point

x = 0

into

2 x^{3} - 21 x^{2} + 60 x \Rightarrow y = 0

Minimum (0, 0)

Plug the extreme point

x = 2

into

2 x^{3} - 21 x^{2} + 60 x \Rightarrow y = 52

Maximum (2, 52)

Plug the extreme point

x = 5

into

2 x^{3} - 21 x^{2} + 60 x \Rightarrow y = 25

Minimum (5, 25)

Plug the extreme point

x = 6

into

2 x^{3} - 21 x^{2} + 60 x \Rightarrow y = 36

Maximum (6, 36)

Minimum (0, 0), Maximum (2, 52), Minimum (5, 25), Maximum (6, 36)

f (x) = e^{x^{2} - 6 x - 1}

f (x) = 4 x^{3} - 50 x^{2} + 150 x

f (x) = - x^{2} + 6 x

f (x) = (x - 1)^{3} (x + 1)^{3}

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

What is the extreme f(x)=2x^3-21x^2+60x,0<= x<= 6 ?
The extreme f(x)=2x^3-21x^2+60x,0<= x<= 6 is Minimum(0,0),Maximum(2,52),Minimum(5,25),Maximum(6,36)