How to find the critical numbers of g ( θ ) = 4 θ - tan ( θ ) ?

Question

How to find the critical numbers of             g              (        θ        )              =    4    θ    -          tan              (        θ        )            ?

tabuaa5q · Accepted Answer

Extrema      g    ′          (      θ      )        =    4    -                  1                                          cos            2                                (            θ            )                                =    0  or                    cos        2                    (        θ        )              =                  1                    4                        cos              (        θ        )              =    ±                  1                    2            That is      θ    =                  π                    3                  θ    =                  5        π                    3            for             cos              (        θ        )              =                  1                    2            and      θ    =                  2        π                    3                  θ    =                  4        π                    3            for             cos              (        θ        )              =    -                  1                    2                  g    ′    ′          (      θ      )        =    -          (      -      2      )                      cos                  -          3                            (        θ        )                    (      -              sin                  (          θ          )                    )        =    -    2                  cos                  -          3                            (        θ        )                    sin              (        θ        )             graph{4x-tan(x) [-7, 7, -7, 7]}  graph{sin(x)/cos(x)^3 [-7, 7, -30, 30]}The second derivative of g, as shown on the bottom plot, changes sign at x = 0, as anticipated. Therefore x=0 is an inflection point. The extrema are true extrema. The second derivative does not change sign at the extrema. If it did, these would not be extrema but saddle points.

How to find the critical numbers of g(theta) = 4theta-tan(theta)?

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