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.