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If the graph of a differentiable function is symmetric

  1. Dec 4, 2011 #1
    1. The problem statement, all variables and given/known data
    If the graph of a differentiable function f is symmertic about the line x=a, what can you say about the symmetry of the graph f'?


    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Dec 4, 2011 #2

    hunt_mat

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    Take a simple example, the function [itex]f(x)=x^{2}[/itex] satisfies the questions criterion. What can you say about the derivative function?
     
  4. Dec 4, 2011 #3
    Well, I was thinking that too. It is only a line, though, so I was confused on what to conclude. Would it be point symmetric at x=a?
     
  5. Dec 4, 2011 #4

    hunt_mat

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    Or take the function [itex]f(x)=x^{4}/4[/itex] as another example and look at the points [itex]x=\pm 2[/itex] for example, what is the value of the derivatives at these points?
     
  6. Dec 4, 2011 #5
    f(2)=4 and f(-2)=-4. So on either side of the x=a the values are negated?
     
  7. Dec 4, 2011 #6
    Or would it be that it is an odd function about x=a?
     
  8. Dec 5, 2011 #7

    HallsofIvy

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    That depends upon what you mean by "an odd function about x= a"!

    If the graph of y= f(x) is symmetric about x= a, then the graph of y= f(x+ a) is symmetric about x= 0- an even function. It follows that y= f'(x+ a) is an odd function- "symmetric through the origin" and so y= f'(x) is "symmetric through (a, 0), not necessarily an "odd function".

    (Note that since f'(x+a) is an odd function, f'(a)= 0.)
     
    Last edited: Dec 5, 2011
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