math- mean value theorem

Hi I am having some trouble with these few quetions I would appreciate some help so that I can understand them better.

1) What, if anything, does the mean value theorem guarantee for the given function on this interval?

a) f(x) = x^2 - 2x + 5 on [1,4]

--I am a bit uncertain on how to answer this, I started out with

f '(c) = f(b) - f(a) / b-a = 13 - 4 / 3 = 9 / 3 = 3.

Then I plugged in 3 for f(x) and got 8.

Does this mean that (3,8) is a critical point?

b) g(x) = 8 / [(x-2)^2] on [1,4]

I am sure this one needs a similiar approach to the last


Lastly:

What values c (if any) are predictable by the mean value theorem for the function f(x) = (x-2)^3 on the interval [0,2]?

I proceeded similiarly here like the last question.

f '(c) turned out to be 4, and f(4) was 8.

I would greatly appreciate some help, since I am having trouble understanding the question and what it is asking.

Thanks!

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  1. f '(c) = f(b) - f(a) / b-a = 13 - 4 / 3 = 9 / 3 = 3.

    There exists a point c in the interval [1,4] such that f'(c) = 3

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  2. ok thanks

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  3. 4+20=46

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  4. verify that the function to satisfies roller's theorem of f /x/=x-3x+2x+5 [0,2].

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