PROBLEM: I wonder why my answer is wrong.
QUESTION: Suppose $x(t)$ satisfies the hypothesis of the Mean value Theorem (MVT), that is, $x(t)$ is continuous on $A \le t \le B$ and differentiable on $A \lt t \lt B$ and $m$ is the lower bound, then: $$m \le x'(t)$$ for all $t$ such that $$A \lt t \lt B$$
Let us consider the MVT on shorter intervals. The strongest conclusion we can draw is:
MVT says it is differentiable over the interval $A \lt t \lt B$ and since differential is $$\frac{x(b)-x(a)}{b-a}$$ therefore $$ m \le \frac{x(b)-x(a)}{b-a}$$ for all $a,b$ over the same interval. Why my answer is wrong ? Why the interval changed to $A \le t \le B$