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For example, if you have a graph $y=x$ and you want to find the values of $c$ that satisfy the mean value theorem for $x\in[1, 3]$, do the points $c=1$ and $c=3$ count as valid? I only ask because for a homework problem $y=-x^3+4x^2-3; [0, 4]$ find values of $c$ that satisfy the MVT, the answer was $0$ and $8/3$ but $0$ was not considered a valid answer because it was on the interval endpoint (to the best of my knowledge).

Jack Pan
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    It just depends on how the theorem is stated. Considering $c$ to be in $(a,b)$ is a stronger and better result, so people often state the MVT that way. In general, you should look for a point in the open interval, as the MVT guarantees that one exists. – florence Aug 07 '16 at 19:12
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    @florence FYI, I converted your perfectly good answer-in-a-comment into a community wiki answer. –  Aug 07 '16 at 19:27

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"It just depends on how the theorem is stated. Considering $c$ to be in $(a,b)$ is a stronger and better result, so people often state the MVT that way. In general, you should look for a point in the open interval, as the MVT guarantees that one exists." – @florence, in a comment