I won't be posting new problems for a couple of weeks. This one is nice:
Suppose and is a function (see here for the definition) and for all Show that is a polynomial of degree at most
if we assume that f(x) is a polynomial of the degree n-1,
we get,
If f(x) has the degree n-2,
then the (n-1)th derivative of f becomes zero.
The same holds for all degrees <(or equal to) (n-1)
hence, the given product always becomes vanishes.
However, if we now start making the degrees > (n-1)
Lets assume that the function is a polynomial of degree n.
I'll take the simplest case here. assuming
Then, the nth derivative of the function:
And all the other derivatives would be of the form:
it follows that
that none of the derivatives will now be zero hence, the product cannot be zero.