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?

        |
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       2*           :
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      - + - - - - - + - -
        |           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: $\displaystyle \,f(3)\;=\;32$