Question

If f(x) has a property that f(2-x) = f(2+x) for all x and f(x) has exactly 4 real zeros, how do you find their sum?

Answer 1

8

Explanation:

Since `f(2-x)=f(2+x)` for all `x`, we must have `f(t) = f(4-t)` for all `t`.

Let `a` be one of the real roots, then `f(a)=f(4-a)=0`, so that `4-a` must be another real zero. (This works even if `a=2`, in which case `4-a=2` is to be counted as another of the real roots,and 2 must be a double root). If `b` is a root different from both `a` and `4-a`, then `4-b` must be the fourth zero (it is easy to check that this is different from either `a` or `4-a`).

So, each pair of zeroes add up to 4, and the sum of all four is 8.