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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 $(a,b)$

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 $(a,b)$