Question

Mean Value Theorem:
It states that F'(c)= (F(b)-F(a))/(b-a) for c on a 14 years ago

Answer 1

it must be differntiable on the open interval \((a,b)\) and continuous on the closed interval \([a,b]\)

Answer 2

Yup! What about "It states that F'(c)= (F(b)-F(a))/(b-a) for c on a<c<b"?

Answer 3

it actually states that given those conditions THERE EXISTS A \(c\in(a,b)\) with
\[f'(c)=\frac{f(b)-f(a)}{b-a}\]

Answer 4

it is an "existance" theorem. says that at least one such number exists

Answer 5

So for example, on a test, if a question asked if it MUST be true that there exists such a point, the answer would be no?

Answer 6

no the answer would be yes, one must exist, provided the hypothesis is met

Answer 7

given a function f that is
a) continuous on the closed interval [a,b] and
b) differentiable on the open interval (a,b)
then there exists some \(c\in (a,b)\) such that \(f'(c)=\frac{f(b)-f(a)}{b-a}\)

Answer 8

Oh, I meant it was one of those questions where it lists three things. For the given function, it is definitely continuous, but not necessarily differentiable.

Answer 9

I think I understand the mean value theorem, I just wanted to make sure about it

Answer 10

but be careful about condition b. differentiable on the open interval doesn't mean "i can find the derivative" it means the derivative exists at every point in the interval

Answer 11

Yup I under stand that

Answer 12

so for example
\[f(x)=\sqrt[3]{x-2}\] has a derivative, but that derivative does not exist at x = 2

Answer 13

Derivative must exist, and the left and right hand limits must be equal, no cusps, corners, something like that right?

Answer 14

right

Answer 15

if you want a good understanding, see if you can convince yourself that the point c guarenteed by the mean value theorem will always be in the center of the interval if your function is quadratic.
check for example that if
\[f(x)=x^2+2x+1\] on the interval \([1,3]\) then the point c will be 2

Answer 16

and if you really want a thorough understanding, work through the proof in your book with a specifice function, say \(f(x)=x^3-3x^2+1\) instead of the generic \(f\) used in the proof to see precisely how it works in a specific example

Answer 17

Okay, thanks for all the help!

Answer 18

yw