ANSWER
Yes, f is continuous on [0, 2] and differentiable on (0, 2) since polynomials are continuous and differentiable on double-struck R.
EXPLANATION
We want to determine whether ,
satisfies the Mean Value Theorem, on the interval,
[0,2].
We check to see if the function satisfies all the hypotheses of the Mean Value Theorem.
1. The function is continuous on the closed interval, [0,2]
2. The function is differentiable on the open interval (0,2).
3. There is a c, such that,
Since the given function is a polynomial it is continuous on [0,2] and differentiable on (0,3) because polynomials are continuous and differentiable on the entire real numbers.
We now differentiate the function to obtain;
This implies that;
This implies that: