1. The Mean Value Theorem (MVT) states that for a function $f$ continuous on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$, there exists at least one point $c$ in $(a,b)$ such that: $$f'(c) = \frac{f(b) - f(a)}{b - a}$$ 2. This means the instantaneous rate of change (the derivative) at some point $c$ equals the average rate of change over $[a,b]$. 3. Important conditions: $f$ must be continuous on $[a,b]$ and differentiable on $(a,b)$. 4. To apply MVT, first verify these conditions. 5. Then compute the average rate of change $\frac{f(b) - f(a)}{b - a}$. 6. Finally, find $c$ such that $f'(c)$ equals this average rate. 7. This theorem helps understand function behavior and guarantees at least one tangent parallel to the secant line between $a$ and $b$.