What is the extreme f(x)=3csc(x) ?

The extreme f(x)=3csc(x) is Minimum(pi/2+2pin,3),Maximum((3pi)/2+2pin,-3)

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extreme f(x)=3csc(x)

extreme points

f (x) = 3 csc (x)

Minimum (\frac{π}{2} + 2 πn, 3), Maximum (\frac{3 π}{2} + 2 πn, - 3)

Solution steps

Find the critical points:

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

Domain of

3 csc (x) : 2 πn < x < π + 2 πn or π + 2 πn < x < 2 π + 2 πn

Find intervals:

Decreasing

: 2 πn < x < \frac{π}{2} + 2 πn,

Increasing

: \frac{π}{2} + 2 πn < x < π + 2 πn,

Increasing

: π + 2 πn < x < \frac{3 π}{2} + 2 πn,

Decreasing

: \frac{3 π}{2} + 2 πn < x < 2 π + 2 πn

Plug the extreme point

x = \frac{π}{2} + 2 πn

into

3 csc (x) \Rightarrow y = 3

Minimum (\frac{π}{2} + 2 πn, 3)

Plug the extreme point

x = \frac{3 π}{2} + 2 πn

into

3 csc (x) \Rightarrow y = - 3

Maximum (\frac{3 π}{2} + 2 πn, - 3)

Minimum (\frac{π}{2} + 2 πn, 3), Maximum (\frac{3 π}{2} + 2 πn, - 3)

f (x) = - x e^{- x} - 4 e^{- x}

f (x) = 20 x^{3} + 9 x^{2} - 24 x + 12

f (x) = x^{2} + 9 x + 14

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

f (x) = \frac{x ^{4}}{4} - \frac{4 x ^{3}}{3} + 2 x^{2} - 1

What is the extreme f(x)=3csc(x) ?
The extreme f(x)=3csc(x) is Minimum(pi/2+2pin,3),Maximum((3pi)/2+2pin,-3)