In this section, Rolle's theorem is introduced along with three applications.

Rolles's theorem is used to find a function's horizontal tangent line. It is a special case of the mean value theorem which is discussed in the next section. Rolle's theorem states:

If a function satisfies the following conditions

• the function is continuous in a closed interval [a, b],
• the function is differentiable on this open interval (a, b),
• the value of the function at the two ends are equal,

then there is a point between a and b such that its derivative is 0.

This theorem can be better understood by examining the following cases:

• First, examine the case when the function f(x) is a constant. In this case, df(x)/dx = 0 at all the points from a to b. For example, when a car is driving at an uniform velocity from time a to b, the derivative of the velocity, acceleration, is 0 in this period.

• Next, take the case when the function value is larger than the starting point value, f(x) > f(a). In this case the function value is maximum, when df(x)/dx equals 0. For example, when a ball is thrown directly upward, velocity is 0 when the displacement is maximum. In other words, the function maximum will be at point c (see graphic) where the tangent is horizontal.

• Another case is when the function value is smaller than the starting value f(x) < f(a). In this case, df(x)/dx = 0 happens at the minimum value point. For example, the derivative of a hanging cable at its lowest point is 0. At point c (see graphic) the function f(x) is a horizontal tangent line.