What is the extreme csc(x) ?

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

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extreme csc(x)

extreme points

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)

f (x) = \frac{x}{x + 6}

f (x) = 3 x^{3} - 9 x + 1

f (x) = - 4 x^{2} + 6 x + 1

f (x) = x^{3} e^{x}

sin (2 x)

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