need help: largest value possible for f(3) on interval

[integragirl]

Suppose that f is continuous on the interval [0,3], f(0) = 2 and 4 < f'(x) < 10 for all x in (0,3). What is the largest that f(3) can possibly be?

 

[royhaas]

Use the mean value theorem.

 

[soroban]

Hello, integragirl!

You need nothing fancy for this problem
\(\displaystyle \;\;\). . . just some Thinking.

Suppose that \(\displaystyle f\) is continuous on the interval [0,3],
\(\displaystyle f(0)\,=\,2\) and \(\displaystyle 4\,\leq\,f'(x)\,\leq\,10\) for all \(\displaystyle x\) in (0,3).
What is the largest that \(\displaystyle f(3)\) can possibly be?

        |
        |   /       :
        |  /        :
        | /         :
        |/          :
       2*           :
        |           : 
      - + - - - - - + - -
        |           3

The graph starts at (0,2) and is increasing.
\(\displaystyle \;\;\)The slope is between +4 and +10.

To reach maximum height at \(\displaystyle x = 3\), the slope should be +10 all the way.

The function would be: \(\displaystyle \,f(x)\:=\:10x\,+\,2\)

Therefore, the maximum value is: \(\displaystyle \,f(3)\;=\;32\)