Does the function satisfy the hypotheses of the Mean Value Theorem on the given interval?

Question

Does the function satisfy the hypotheses of the Mean Value Theorem on the given interval?

f(x) = 2x² − 3x + 1,    [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     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).

c =

Answer 1

f'(c) = (f(b) - f(a))/(b - a); f(x) = 2x² - 3x + 1; f'(x) = 4x - 3; f'(c) = 4c - 3; a = 0; b = 2; f(0) = 1; f(2) = 8 - 6 + 1 = 3;

(f(2) - f(0))/(2 - 0) = (3 - 1)/2 = 1; 4c - 3 = 1; 4c = 4; c = 1