Help me to solve this problem please? It's about real analysis. I am studying for preparing my midterm, but I still have many difficulties.

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1 Answer
Apr 1, 2018

Let #f(x)# be derivable for every #x in RR# and be an odd function, so that:

#f(-x) = -f(x)#

Differentiating both sides of the identity we have:

#-f'(-x) = -f'(x)#

or:

#f'(-x) = f'(x)#

which means that #f'(x)# is even.

Differentiating again:

#f''(-x) = -f''(x)#

which means #f''(x)# is odd, and so forth, as long as #f^((n))(x)# is derivable.

The easier examples are polynomials: a polynomial containing only odd order powers:

#P(x) = sum_(k=0)^N a_k x^(2k+1)#

is an odd function, while its derivative;

#P'(x) = sum_(k=0)^N a_k(2k+1) x^(2k)#

contains only even ordered powers and is an even function.

For instance see below the graph of:

#P(x) = 3x^5-4x^3+2x#

#P'(x) = 15x^4-12x^2+2#
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