# Find the critical numbers of the function on the interval 0 ≤ θ < 2π. g(θ) = 4 θ - tan(θ)

By definition, critical numbers of a function `g` are the values of argument at which `g '` is zero or does not exist.

The given function `g ( theta ) = 4 theta - tan ( theta )` itself is defined everywhere except `theta = pi / 2 + n...

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By definition, critical numbers of a function `g` are the values of argument at which `g '` is zero or does not exist.

The given function `g ( theta ) = 4 theta - tan ( theta )` itself is defined everywhere except `theta = pi / 2 + n pi , n in ZZ.`

Its first derivative is `g ' ( theta ) = 4 - 1 / cos ^ 2 ( theta ),` which also exists everywhere except `theta = pi / 2 + n pi , n in ZZ.` Where does it equal zero?

`4 - 1 / cos ^ 2 ( theta ) = 0` is the same as `cos ^ 2 ( theta ) = 1 / 4,` which is the same as `cos ( theta ) = +- 1 / 2 .` Such points are `+- pi / 3 + 2 n pi, n in ZZ` and `+- ( 2 pi ) / 3 + 2 n pi , n in ZZ.`

The last task is to determine which of these points belong to the given interval (its inner part `( 0 , 2 pi )`). They are `pi / 3 , ( 2 pi ) / 3 , ( 4 pi ) / 3 , ( 5 pi ) / 3.`

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