Does the function satisfy the hypotheses of the mean 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 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). c =

The true statement is: (b) Yes, f is continuous on [0, 2] and differentiable on (0, 2) since polynomials are continuous and differentiable

Mean value theorem states that:

If [a,b] and

on (a,b),

Then there is a point c in (a,b), such that:

The function is given as:

And the interval is:

We have

This becomes

Calculate f(2) and f(0)

So, we have:

Recall that:

Differentiate

Substitute c for x

Substitute 2 for f'(c)

Collect like terms

Divide both sides by 6

The interval is given as: [0,2]

The value of c is true for interval (0,2).

Hence, the true statement is:

(b) Yes, f is continuous on [0, 2] and differentiable on (0, 2) since polynomials are continuous and differentiable

Read more about mean value theorems at:

link

is a polynomial and thus continuous everywhere and differentiable on any open interval. (second option)

The MVT then guarantees the existence of such that

We have

so

i think the answer is seven, for if i'm being useless °ω°

step-by-step explanation: