Find antiderivative with given conditionLet f : R → R be the function defined by f ( x ) = max { 1 , x 2 } .. Prove that f admits antiderivatives on R and find one antiderivative F such that 4 F ( − 3 2 ) − 3 F ( 1 2 ) = 3 F ( 2 )I have used the formula with modulus to rewrite the function but I cannot think of a function whose derivative is a modulus function. How should I proceed?

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Find antiderivative with given conditionLet   f  :      R    →      R   be the function defined by   f  (  x  )  =  max  {  1  ,      x    2    }  .. Prove that f admits antiderivatives on       R   and find one antiderivative F such that   4  F      (    −          3      2        )    −  3  F      (          1      2        )    =  3  F  (  2  )I have used the formula with modulus to rewrite the function but I cannot think of a function whose derivative is a modulus function. How should I proceed?

svirajueh · Accepted Answer

Step 1Since   f  (  x  )  =      {                                        x            2                                     if           x          ⩽          −          1                                      1                           if           −          1          ⩽          x          ⩽          1                                                  x            2                                     if           x          ⩾          1          ,                        .an antiderivative a of f is   a  (  x  )  =      {                                        1            3                                x            3                    −                      2            3                                     if           x          ⩽          −          1                                      x                           if           −          1          ⩽          x          ⩽          1                                                  1            3                                x            3                    +                      2            3                                     if           x          ⩾          1.                        Step 2So, every primitive F of f is of the form   a  +  k for some constant k. An easy computation shows that the only k for which the equality   4  F      (    −          3      2        )    −  3  F      (          1      2        )    =  3  F  (  2  ) holds is   k  =  −            28      3

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