If the graph of a differentiable function is symmetric

[NWeid1]

Homework Statement

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'?

Homework Equations

The Attempt at a Solution

 

[hunt_mat]

Take a simple example, the function $f(x)=x^{2}$ satisfies the questions criterion. What can you say about the derivative function?

 

[NWeid1]

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?

 

[hunt_mat]

Or take the function $f(x)=x^{4}/4$ as another example and look at the points $x=\pm 2$ for example, what is the value of the derivatives at these points?

 

[NWeid1]

f(2)=4 and f(-2)=-4. So on either side of the x=a the values are negated?

 

[NWeid1]

Or would it be that it is an odd function about x=a?

 

[HallsofIvy]

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.)