What is the extreme 2cos(3x)+2sin(10)y+3x-10y=0 ?

The extreme 2cos(3x)+2sin(10)y+3x-10y=0 is

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extreme 2cos(3x)+2sin(10)y+3x-10y=0

extreme

Maximum (\frac{π}{18} + \frac{2 πn}{3}, \frac{- 12 cos ( \frac{π}{6} + 2 πn ) - π - 12 πn}{12 ( - 5 + sin ( \frac{π}{18} ) )}), Minimum (\frac{5 π}{18} + \frac{2 πn}{3}, \frac{- 12 cos ( \frac{5 π}{6} + 2 πn ) - 5 π - 12 πn}{12 ( - 5 + sin ( \frac{π}{18} ) )})

Solution steps

Convert

10

degrees to radians:

\frac{π}{18}

\frac{- 2 cos ( 3 x ) - 3 x}{2 ( - 5 + sin ( \frac{π}{18} ) )}

Find the critical points:

x = \frac{π}{18} + \frac{2 πn}{3}, x = \frac{5 π}{18} + \frac{2 πn}{3}

Domain of

\frac{- 2 cos ( 3 x ) - 3 x}{2 ( - 5 + sin ( \frac{π}{18} ) )} : - \infty < x < \infty

Find intervals:

Increasing

: \frac{2 π}{3} n \leq x < \frac{π}{18} + \frac{2 πn}{3},

Decreasing

: \frac{π}{18} + \frac{2 πn}{3} < x < \frac{5 π}{18} + \frac{2 πn}{3},

Increasing

: \frac{5 π}{18} + \frac{2 πn}{3} < x < \frac{2 π}{3} + \frac{2 π}{3} n

Plug the extreme point

x = \frac{π}{18} + \frac{2 πn}{3}

into

\frac{- 2 cos ( 3 x ) - 3 x}{2 ( - 5 + sin ( \frac{π}{18} ) )} \Rightarrow y = \frac{- 12 cos ( \frac{π}{6} + 2 πn ) - π - 12 πn}{12 ( - 5 + sin ( \frac{π}{18} ) )}

Maximum (\frac{π}{18} + \frac{2 πn}{3}, \frac{- 12 cos ( \frac{π}{6} + 2 πn ) - π - 12 πn}{12 ( - 5 + sin ( \frac{π}{18} ) )})

Plug the extreme point

x = \frac{5 π}{18} + \frac{2 πn}{3}

into

\frac{- 2 cos ( 3 x ) - 3 x}{2 ( - 5 + sin ( \frac{π}{18} ) )} \Rightarrow y = \frac{- 12 cos ( \frac{5 π}{6} + 2 πn ) - 5 π - 12 πn}{12 ( - 5 + sin ( \frac{π}{18} ) )}

Minimum (\frac{5 π}{18} + \frac{2 πn}{3}, \frac{- 12 cos ( \frac{5 π}{6} + 2 πn ) - 5 π - 12 πn}{12 ( - 5 + sin ( \frac{π}{18} ) )})

Maximum (\frac{π}{18} + \frac{2 πn}{3}, \frac{- 12 cos ( \frac{π}{6} + 2 πn ) - π - 12 πn}{12 ( - 5 + sin ( \frac{π}{18} ) )}), Minimum (\frac{5 π}{18} + \frac{2 πn}{3}, \frac{- 12 cos ( \frac{5 π}{6} + 2 πn ) - 5 π - 12 πn}{12 ( - 5 + sin ( \frac{π}{18} ) )})

What is the extreme 2cos(3x)+2sin(10)y+3x-10y=0 ?
The extreme 2cos(3x)+2sin(10)y+3x-10y=0 is