Rolle’s Theorem

In this section we shall review the Rolle’s theorem. The theorem is named after the seventeenth century French mathematician Michel Rolle (1652-1719).
(Rolle’s Theorem) : Let f be a continuous function defined on [a,b] and differentiable on ]a,b[. If f(a)=f(b), then there exists a number xo in ]a,b[ such that f,(x)=0
Geometrically, we can interpret the theorem easily. You know that since f is continuous graph of f is a smooth curve
Definition of Rolle's Theorem
As we that the derivative f′(x) at some point xo gives the slope of the tangent at (xo,f(xo)) to the curve y=f(x). Therefore the theorem states that if the end values f(a) and f(b) are equal, then there exists a point xo in ]a,b[ such that the slope of the tangent at the point P(xo,f(xo)) is zero, that is, the tangent is parallel to x−axis at that point. In fact we can have more than one point at which f′(x)=0 as shown in Fig. This shows that the number xo in Theorem may not be unique.
The following example gives an application of Rolle’s theorem.
Example : Use Rolle’s theorem to show that there is a solution of the equation cotx=x in ]0,π2[ Solution : Here we have to solve the equation cotx−x=0. We rewrite cotx−x as cotx−xsinxsinx. Solving the equation cotx−xsinxsinx=0 in ]0,π2[ is same as solving the equation cotx−xsinx=0. Now we shall see whether we can find a function f which satisfies the conditions of Rolle’s theorem and for which f′(x)=cotx−xsinx. Our experience in differentiation suggests that we try f(x)=xcosx. This function f is continuous in [0,π2] differentiable in ]0,π2[ and the derivative f′(x)=cotx−xsinx. Also f(0)=0=f(π2). Thus f satisfies all the requirements of Rolle’s theorem. Hence, there exists a point xo, in ]0,π2[ such that f′(xo)=cotxo−xosinxo=0. This shows that a solution to the equation cotx−x=0 exists in ]0,π2[