Question

Suppose that f(x) is continuous on [2,4] and f '(x) exists on (2,4) and f' (c) = (f(4) - f(2)) / 2, then C is in (2,4). True or False?

Answer 1

mean value theorem

Answer 2

i know can you help me use it.

Answer 3

hmm - i'm very rusty at that im afraid

Answer 4

sorry

Answer 5

can you state the mean value theorem?

Answer 6

f(b)-f(a) / b-a ?

Answer 7

hello ?

Answer 8

T

Answer 9

what does the mean value theorem say about
f'(c) for some c in (a, b) ?

Answer 10

it says that "if f is differentiable on an interval (a, b), then there exists some c such that \[f'(c)={f(a)-f(b)\over a-b}\]

Answer 11

... where c is in (a,b)

Answer 12

Geometric interpretation of this problem:

{ f(4) - f(2) } / (4 - 2) is the slope of the line AB.

There exists a point D in the interval (2, 4) where the slope of the tangent at D is parallel to AB.

Answer 13

Slope of the tangent at D = f'(c)

Answer 14

to be exact, that's a geometric interpretation of the mean value theorem. The problem asks if the converse is true: i.e. "if the slope of D=f'(c) is the same as the line AB, is c in (a,b) ?"

Answer 15

Then the answer is "Not always true". There will definitely be at least one c in (a,b). But there can be a c outside of (a,b) too. And that c will not be in (a,b).