Maximize this parametric functionFor what values of a &gt; 0 is 1 1 + | x | + 1 1 + | x − a | is maximized?

Question

Maximize this parametric functionFor what values of   a  &amp;gt;  0 is      1          1      +      |      x      |        +      1          1      +      |      x      −      a      |      is maximized?

ejigaboo8y · Accepted Answer

Step 1Set  f  (  x  )  =      1          1      +      |      x      |        +      1          1      +      |      x      −      a      |      And split it in the following way  f  (  x  )  =      {                                        1                          1              −              x                                +                      1                          1              −              x              +              a                                                            if                           x              &amp;lt;              0                                                                        1                          1              +              x                                +                      1                          1              −              x              +              a                                                            if                           0              ≤              x              &amp;lt;              a                                                                        1                          1              +              x                                +                      1                          1              +              x              −              a                                                            if                           x              ≥              a                                .                        and calculating the first derivative you obtain      f    ′    (  x  )  =      {                                        1                          (              a              −              x              +              1                              )                2                                              +                      1                          (              1              −              x                              )                2                                                                          if                           x              &amp;lt;              0                                                                        1                          (              a              −              x              +              1                              )                2                                              −                      1                          (              x              +              1                              )                2                                                                          if                           0              &amp;lt;              x              &amp;lt;              a                                                            −                      1                          (              −              a              +              x              +              1                              )                2                                              −                      1                          (              x              +              1                              )                2                                                                          if                           x              &amp;gt;              a                                .                        You can immediately notice that f(x) is increasing for   x  &amp;lt;  0 since      {          1              (        a        −        x        +        1                  )          2                      +          1              (        1        −        x                  )          2                      &amp;gt;    0                      |                  ∀    x    ∈          R        }    .Also,      1          (      a      −      x      +      1              )        2              −      1          (      x      +      1              )        2              =            −              a        2            +      2      a      x      −      2      a      +      4      x              (      x      +      1              )        2            (      a      −      x      +      1              )        2            which is negative for   x  &amp;lt;  a      /    2 , therefore the function f(x) is decreasing in       (    0    ,          a      2        )   . This implies that   x  =  0 is a maximum. Also, being that f(x) is symmetric with respect to   x  =  a      /    2 , there is another maximum point, with the same value of the one found previously, at   x  =  a . The function at these critical points has value:  f  (  0  )  =  f  (  a  )  =            2      +      a              1      +      a        .

Maximize this parametric function For what values of a > 0 is 1

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