c. What is the degree of f(k)(x) where k is any integer greater than n?
Suppose f (x) is a degree n polynomial.
a. The degree of f(n - 1)(x) is 1. After taking (n- 1) derivatives of a degree n polynomial, we're left with something of degree 1.
b. The degree off(n)(x) is 0. After taking n derivatives of a degree n polynomial we're left with a constant, which has degree 0.
c. The degree of f(k)(x) , where k is any integer greater than n, is 0. Since f(n) is a constant (in other words, has degree 0), any subsequent derivative will be the derivative of a constant,
and therefore zero.