What is the critical 1/2 csc(3x) ?

The critical 1/2 csc(3x) is x= pi/6+(2pi)/3 n,x= pi/2+(2pi)/3 n

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critical 1/2 csc(3x)

critical points

\frac{1}{2} csc (3 x)

x = \frac{π}{6} + \frac{2 π}{3} n, x = \frac{π}{2} + \frac{2 π}{3} n

Solution steps

\frac{1}{2} csc (3 x)

Find where

f^{'} (x)

is equal to zero or undefined

x = \frac{π}{6} + \frac{2 π}{3} n, x = \frac{π}{2} + \frac{2 π}{3} n, x = \frac{2 π}{3} n, x = \frac{π}{3} + \frac{2 π}{3} n

Identify critical points not in

f (x)

domain

Domain of

\frac{1}{2} csc (3 x) : \frac{2 π}{3} n < x < \frac{π}{3} + \frac{2 π}{3} n or \frac{π}{3} + \frac{2 π}{3} n < x < \frac{2 π}{3} + \frac{2 π}{3} n

\frac{1}{2} csc (3 x)

is not defined at

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

, therefore:

x = \frac{π}{6} + \frac{2 π}{3} n, x = \frac{π}{2} + \frac{2 π}{3} n

f (x) = 6 x^{4} - 4 x^{6}

f (x, y) = y x - y^{2} - 5 x + 19 y

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

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

t^{3} - 9 x^{2} + 27 x

What is the critical 1/2 csc(3x) ?
The critical 1/2 csc(3x) is x= pi/6+(2pi)/3 n,x= pi/2+(2pi)/3 n