Continuous?

Example:   

 

f(x) = x

 

(f(4) - f(2))/2 = 1

 

But the slope of f(x) is 1 everywhere, so for any c the value of f'(c) will equal (f(4) - f(2))/2

 

Melody is correct.

.

I would like another mathematician to read what I have put and tell me if they agree that it is entirely right and if it is not then tell me why.

 

True or False. If f is continuous on [2,4], and f1(c)= (f(4)-f(2))/2,  then c ∈ [2,4].

 

I think that you mean:

True or False. If f is continuous on [2,4], and f'(c)= (f(4)-f(2))/2,  then c ∈ [2,4].

 

(f(4)-f(2))/2 is the gradient of the line joining the points (2,f(2)) and (4,f(4))

so yes of course, there must be a point on f(x) between 2 and 4 where the gradient of the tangent is equal to that.

So 

yes 

There is at least one c  between [2,4] that will make that true.     There could be other c's elsewhere that make it true too.

 

If you draw a pic and think about it you might realize why I think the answer is obvious  (I hope I am entirely right)