Maximum of 1 + 3 a 2 ( a 2 + 1 ) 2 What is the maximum value of 1 + 3 a 2 ( a 2 + 1 ) 2 , given that a is a real number, and for what values of a does it occur ?.

Question

Maximum of             1      +      3              a        2                    (              a        2            +      1              )        2            What is the maximum value of                                           1            +            3                          a                              2                                                                        (                              a                                  2                                            +              1              )                                      2                                          , given that   a is a real number, and for what values of   a does it occur ?.

Jakob Chavez · Accepted Answer

Writing it as                                                   1          +          3                      a            2                                                                            (                      a            2                    +          1                      )            2                                =                                                  3          (                      a            2                    +          1          )          −          2                                                                  (                      a            2                    +          1                      )            2                                =                                                  3                                                                              a            2                    +          1                      −                                                  2                                                                  (                      a            2                    +          1                      )            2                                 gives a quadratic in   x  =                                                  1                                                                              a            2                    +          1                      , with a maximum at   x  =                                                  3                                                                  4                       (    ⟺        a    2    =  1      /    3  ) of value                                                   3          ⋅          3                                                                  4                      −                                                  2          ⋅          9                                                                  16                      =                                                  9                                                                  8

ferdysy9 · Accepted Answer

Let   x  =      a    2    ≥  0. We then have to look at            1      +      3      x              (      x      +      1              )        2            upon differentiating and setting equal to   0, we get            3      (      x      +      1              )        2            −      2      (      x      +      1      )      (      1      +      3      x      )              (      x      +      1              )        4                  ⟹    3  (  x  +  1      )    2    =  2  (  x  +  1  )  (  1  +  3  x  )      ⟹    3  (  x  +  1  )  =  2  (  1  +  3  x  )      ⟹    x  =      1    3  Thus, the relative maxima (or minima) occurs at  a  =  ±      x    =  ±            3        3

What is the maximum value of (1+3a^2)/((a^2+1)^2) , given that a is a real number, and for what values of a does it occur ?.

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