Does a Function Exist Given Specific Function Values and Derivative Constraints?

[Loppyfoot]

Homework Statement

Does the function f exist when f(0)=-1, f(2)=4, and f'(x)≤2 for all values of x? Justify your answer.

The Attempt at a Solution

I am having some trouble with where to start...and end. Thanks for your help.

 

[Dick]

What would the mean value theorem tell you about f(x) on the interval [0,2]?

 

[Loppyfoot]

What would the mean value theorem tell you about f(x) on the interval [0,2]?
It tells me that there exists a number c in (0,2) such that f'(c) = (f(2)-f(0))/(2-0)

 

[Dick]

Ok. What is (f(2)-f(0))/(2-0) as a number?

 

[Loppyfoot]

so 2.5 is not less than or equal to 2, so the function would not exist?

 

[Dick]

so 2.5 is not less than or equal to 2, so the function would not exist?

Exactly.

 

[Loppyfoot]

Great. Thank you, Sir.