Formal proof of h ( x , y ) = f ( x ) + g ( y ) has a critical point ( x 0 , y 0 ) iff x 0 is a critical point of f and y 0 is a critical point of g

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Formal proof of   h  (  x  ,  y  )  =  f  (  x  )  +  g  (  y  ) has a critical point   (      x    0    ,      y    0    ) iff       x    0   is a critical point of f and       y    0   is a critical point of   g

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A critical point   (      x    0    ,      y    0    ) of a function defined everyone on             R        2   is a point where   ∇  f  =  0, i.e.       h    x    (      x    0    ,      y    0    )  =      h    y    (      x    0    ,      y    0    )  =  0 (using subscripts for partial derivatives). Computing the partials, we have      h    x    =      f    x          h    y    =      g    y  so       h    x    (      x    0    ,      y    0    )  =  0 iff       f    ′    (      x    0    )  =  0 and       h    y    (      x    0    ,      y    0    )  =  0 iff       g    ′    (      y    0    )  =  0, in other words,   ∇  h  =  0 iff       x    0   is a critical point of   f and       y    0   is a critical point of   g.

Formal proof of h(x,y)=f(x)+g(y) has a critical point (x_0,y_0) iff x_0 is a critical point of f and y_0 is a critical point of g

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