Problem: If f(x) is a real valued function of the real variable x, and f(x) is not identically zero, and for all a and b,
f(a+b)+f(a−b)=2f(a)+2f(b),
then for all x and y
Answer Choices:
A. f(0)=1
B. f(−x)=−f(x)
C. f(−x)=f(x)
D. f(x+y)=f(x)+f(y)
E. there is a positive number T such that f(x+T)=f(x)
Solution:
Choosing a=b=0 yields
2f(0)f(0)=4f(0)=0
Choosing a=0 and b=−x yields
f(x)+f(−x)f(x)=2f(0)+2f(−x)=f(−x)
Note: A continuous function f satisfies the functional equation if and only if f(x)=cx2 for some fixed number c and for all x.