Real life application of Rolle's theorem
Let's apply Rolle's theorem to the position function s = f(t), where f(t) is continuous and differentiable along its path. The function f(t) relates to a bullet fired from a war tank vertically upward with an initial velocity 40 m/sec.

After it is released from the tank at t = 0, it would reach the same place at t = 8 seconds. Therefore, f(0) = f(8). The bullet reaches its maximum height at t = 4 where the vertical velocity, that is, f '(t) becomes zero which is in the interval (0, 8).
Rolle's Theorem and Mean Value Theorem

**Rolle's theorem: **It was first published in 1691 by the French mathematician **Michel Rolle.** This theorem states that: if a function **f** is continuous on the closed interval [a, b], differentiable on the open interval (a, b), and if f(a) = f(b), then there exists at least one real number 'c' in the open interval (a, b) such that (c) = 0.

From Rolle's theorem, we can see that if a function **f** is continuous on [a, b] and differentiable on (a, b), and if f(a) = f(b), there must be at least one x-value between a and b at which the graph of **f** has a horizontal tangent, as shown in above figure.

**Mean value theorem: **It was first stated by another French mathematician, **Joseph-Louis Lagrange.** This 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 number 'c' in (a, b) such that .

**Geometrical interpretation of the mean value theorem: **Geometrically, the theorem says that somewhere between points A and B on a differentiable curve, there is at least one tangent line parallel to secant line AB.

**Physical interpretation of the mean value theorem: ** If we think of as the average rate of change of the function **f** over the interval [a, b] and (c) as an instantaneous rate of change, then the mean value theorem says that there must be a point in the open interval (a, b) at which the instantaneous rate of change is equal to the average rate of change over the interval [a, b].