Let f ( x , y ) : R × R ↦ R be a continuous and differentiable function. When can we claim that the following holds true: d d x max y ∈ R f ( x , y ) = max y ∈ R ∂ ∂ x f ( x , y )assuming that both max y ∈ R f ( x , y ) and max y ∈ R ∂ ∂ x f ( x , y ) exists.

Question

Let   f  (  x  ,  y  )  :      R    ×          R        ↦          R       be a continuous and differentiable function. When can we claim that the following holds true:      d          d      x            max          y      ∈              R              f  (  x  ,  y  )  =      max          y      ∈              R                  ∂          ∂      x        f  (  x  ,  y  )assuming that both       max          y      ∈              R              f  (  x  ,  y  ) and       max          y      ∈              R                  ∂          ∂      x        f  (  x  ,  y  ) exists.

Mollie Roberts · Accepted Answer

Suppose both maxima occur at the same location, that is, there is a function   g  (  x  ) such that                    f                              (                          x        ,        g        (        x        )                              )                                              =                  max                      y            ∈                          R                                      f        (        x        ,        y        )        ,                                      f          x                                      (                          x        ,        g        (        x        )                              )                                              =                  max                      y            ∈                          R                                                f          x                (        x        ,        y        )        ,            where subscripts denote partial differentiation. Assuming differentiability in y, this implies that             (        x  ,  g  (  x  )            )       is a zero of both       f    y    (  x  ,  y  ) and       f          x      y        (  x  ,  y  ). Then                                          d                                              d                        x                                    max                      y            ∈                          R                                      f        (        x        ,        y        )                            =                              d                                              d                        x                          f                              (                          x        ,        g        (        x        )                              )                                                            =                  f          x                                      (                          x        ,        g        (        x        )                              )                          +                  f          y                                      (                          x        ,        g        (        x        )                              )                                    g          ′                (        x        )                                          =                  max                      y            ∈                          R                                                f          x                (        x        ,        y        )            because       f    y              (        x  ,  g  (  x  )            )        =  0.Thus, if both maxima coincide, then your claim holds. If not, I&#039;d guess all bets are off.

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