f(x) = 3x2 − 5x + 2, [0, 2] , solving of the Mean Value Theorem

Question

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

f(x) = 3x² − 5x + 2,    [0, 2]

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

Yes, f(x) is a polynomial, so it will be continuous and differentiable on any interval (a,b), [a,b], [a,b), or (a,b].

Now we can proceed to find all numbers c such that f'(c)= (f(b) - f(a))/(b - a) on [0, 2] where a=0 and b=2.

First, let's find f'(c):

f'(x) = 6x - 5, so f'(c) = 6c - 5

Now, let's calculate (f(b) - f(a))/(b - a):

(f(b) - f(a)) = (f(2) - f(0)) = (3(2^2) - 5(2) + 2) - (3(0)^2 - 5(0) + 2) = 4 - 2 = 2

b - a = 2 - 0 = 2

So (f(b) - f(a))/(b - a) = 2/2 = 1

Setting f'(c) equal to 1:

6c - 5 = 1

6c = 6

c = 1