Suppose that f(0)=3 and f′(x)≤7 for all values of x. use the mean value theorem to determine how large f(4) can possibly be.

With these given values:

 f(0)=4 and f′(x)≤4 

The solution is as follows:

 f(0) = 4 
f ' (x) ≤ 4 for all x 

The Mean Value Theorem says there is a c between 0 and 4 such that f ' (c) = [f(4) - f(0)] / (4 - 0) 
But since f ' (x) ≤ 4 for all x, then [f(4) - f(0)] / (4 - 0) ≤ 4 
Since f(0) = 4, then [f(4) - 4] / (4 - 0) ≤ 4 
Multiplying both sides by 4 you get f(4) - 4 ≤ 16 
Adding 4 to both sides you get f(4) ≤ 20 

So f(4) ≤ 20

By examining the solution, it could guide you on answering the problem on your own. Hope that helps