What is the extreme f(x)=2x^3-21x^2+60x,1<= x<= 6 ?

The extreme f(x)=2x^3-21x^2+60x,1<= x<= 6 is Minimum(1,41),Maximum(2,52),Minimum(5,25),Maximum(6,36)

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extreme f(x)=2x^3-21x^2+60x,1<= x<= 6

extreme points

f (x) = 2 x^{3} - 21 x^{2} + 60 x, 1 \leq x \leq 6

Minimum (1, 41), Maximum (2, 52), Minimum (5, 25), Maximum (6, 36)

Solution steps

Find the critical points:

x = 1, x = 2, x = 5, x = 6

Domain of

2 x^{3} - 21 x^{2} + 60 x, 1 \leq x \leq 6 : 1 \leq x \leq 6

Find intervals:

Increasing

: 1 < x < 2,

Decreasing

: 2 < x < 5,

Increasing

: 5 < x < 6

Plug the extreme point

x = 1

into

2 x^{3} - 21 x^{2} + 60 x \Rightarrow y = 41

Minimum (1, 41)

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 (1, 41), Maximum (2, 52), Minimum (5, 25), Maximum (6, 36)

y^{3} + 6 x^{2} y - 6 x^{2} - 6 y^{2} + 3

T (x, y) = 100 + x (y + 1)^{2} - 6 x^{2} - 8 (y + 1)^{2}

\frac{( x ^{3} )}{( x - 1 ) ^{2}}

f (x) = x^{\frac{1}{7}} + 3

f (x) = ln (x^{2} + 10 x + 14)

What is the extreme f(x)=2x^3-21x^2+60x,1<= x<= 6 ?
The extreme f(x)=2x^3-21x^2+60x,1<= x<= 6 is Minimum(1,41),Maximum(2,52),Minimum(5,25),Maximum(6,36)