Does the function satisfy the hypotheses of the mean value theorem on the given interval? f(x) = 4x2 − 2x + 3, [0, 2] yes, it does not matter if f is continuous or differentiable, every function satifies the mean value theorem. yes, f is continuous on [0, 2] and differentiable on (0, 2) since polynomials are continuous and differentiable on double-struck r. no, f is not continuous on [0, 2]. no, f is continuous on [0, 2] but not differentiable on (0, 2). there is not enough information to verify if this function satifies the mean value theorem. if it satisfies the hypotheses, find all numbers c that satisfy the conclusion of the mean value theorem. (enter your answers as a comma-separated list. if it does not satisify the hypotheses, enter dne).

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:

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