Show that there is a number x ∈ [ π / 2 , π ] such that tan ⁡ ( x ) = − xCan I solve this by the intermediate value theorem?

Question

Show that there is a number   x  ∈  [  π      /    2  ,  π  ] such that   tan  ⁡  (  x  )  =  −  xCan I solve this by the intermediate value theorem?

Wayne Everett · Accepted Answer

Yes, you can. Consider   f  (  x  )  =  tan  ⁡  x  +  x. Notice that   f  (  π      /    2  +  ϵ  )  ≈  −  ∞ for small   ϵ (more formally, the right limit as   x  →  π      /    2 of   f  (  x  ) is   −  ∞) while   f  (  π  )  =  π. Now the intermediate value theorem gives you want you want.

Donna Flynn · Accepted Answer

Your problem is equivalent to proving that                    (1)                    f        (        z        )        =        z        cot        ⁡        z        =                  z          2                +                              π            z                    2                =        g        (        z        )            for some   z  ∈  I  =      (    0    ,          π      2        )  . That is trivial since   f  (  z  ) decreases from 1 to 0 on   I while   g  (  z  ) increases from 0 to             π      2        2   on   I, and both   f  (  z  ) and   g  (  z  ) are continuous over   I. By approximating   z  cot  ⁡  z with   1  −            z      2        3   we also get:  z  ≈            −      3      π      +              192        +        9                  π          2                      16  hence the solution to the original problem is about:                    (2)                    x        ≈                              5            π            +                          192              +              9                              π                2                                              16                .

Show that there is a number x in [pi/2,pi] such that tan(x)=−x

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