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

The extreme f(x)=csc(x) is Minimum(pi/2+2pin,1),Maximum((3pi)/2+2pin,-1)

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extreme f(x)=csc(x)

extreme points

f (x) = csc (x)

Minimum (\frac{π}{2} + 2 πn, 1), Maximum (\frac{3 π}{2} + 2 πn, - 1)

Solution steps

Find the critical points:

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

Domain of

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

csc (x) \Rightarrow y = 1

Minimum (\frac{π}{2} + 2 πn, 1)

Plug the extreme point

x = \frac{3 π}{2} + 2 πn

into

csc (x) \Rightarrow y = - 1

Maximum (\frac{3 π}{2} + 2 πn, - 1)

Minimum (\frac{π}{2} + 2 πn, 1), Maximum (\frac{3 π}{2} + 2 πn, - 1)

What is the extreme f(x)=csc(x) ?
The extreme f(x)=csc(x) is Minimum(pi/2+2pin,1),Maximum((3pi)/2+2pin,-1)