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The hypotheses ensure that the function behaves smoothly enough for an interior tangent to match the secant slope. A sharp interior corner or a jump at an endpoint can remove this guarantee.

Conditions: The function fails to be continuous on $[a,b]$ or differentiable on $(a,b)$.

The hypotheses ensure that the function behaves smoothly enough for an interior tangent to match the secant slope. A sharp interior corner or a jump at an endpoint can remove this guarantee.

Conditions: The function fails to be continuous on $[a,b]$ or differentiable on $(a,b)$.

The mean value theorem states that if a function $f$ is continuous on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$, then there exists at least one point $c$ in $(a,b)$ such that the instantaneous rate of change (derivative) at $c$ equals the average rate of change over the interval. Mathematically, this is expressed as $f'(c) = \frac{f(b)-f(a)}{b-a}$.

Conditions: $a < b$; $f$ is continuous on $[a,b]$; $f$ is differentiable on $(a,b)$

The mean value theorem states that if a function $f$ is continuous on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$, then there exists at least one point $c$ in $(a,b)$ such that the instantaneous rate of change (derivative) at $c$ equals the average rate of change over the interval. Mathematically, this is expressed as $f'(c) = \frac{f(b)-f(a)}{b-a}$.

Conditions: $a < b$; $f$ is continuous on $[a,b]$; $f$ is differentiable on $(a,b)$