How does Rolle's theorem relate to the mean value theorem?

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 $(a,b)$.

No, the point $c$ is not necessarily unique. The theorem guarantees the existence of at least one such point in the open interval $(a,b)$, but there may be multiple points where the instantaneous rate of change equals the average rate of change.

Conditions: $f$ 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 $(a,b)$

Geometrically, the mean value theorem states that if a function is smooth over an interval, there is at least one point inside the interval where the tangent line is parallel to the secant line connecting the endpoints of the interval. The secant represents the average rate of change, and the tangent represents the instantaneous rate of change.

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