Does the function satisfy the hypotheses of the Man Value Theorem on the given i

Question

Does the function satisfy the hypotheses of the Man Value Theorem on the given interval? f(x)=3x2-4x+1, [0,2] Yes, it does not matter if f is continuous or differentiable, every function satisfies the Mean Value Theorem. Yes, f is continuous on [0,2] and differentiable on (0,2) since polynomials are continuous and differentiable on R. No, f is not continuous on [0,2]. No, f is continuous on [0,2] but nit differentiable on (0,2). There is not enough information to verify if this function satisfies the Mean Value Theorem. If it satisfies the hypotheses, find all numbers c that satisfy the conclusions of the Mean Value Theorem. (Enter your answer as a common-separated list. If it does not satisfy the hypothesis, enter DNE)

Explanation / Answer

by mean value theorem, f'(c) = [f(b) - f(a)]/(b-a) f(b) = f(2) = 3*4 - 4*2 + 1 = 5 f(a) = f(0) = 0 - 0 + 1 = 1 f'(x) = 6x - 4 putting the values in formula 6c - 4 = [5 - 1]/(2-0) 6c - 4 = 2 6c = 6 c = 1............this value lies in [0,2] hence MVT is verified for c = 1

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