5. 8 The Meanvalue Theorem
For the theoretical purpose, the meanvalue theorem is one of the most useful tools. The special and easy case of mean value theorem is the ‘Rolle's’ theorem, which will serve as a lemma for our main results.
Rolle’s Theorem : If ‘f’ is continuous on
[a, b] and differentiable in (a, b) and if, further f (b) = f (a),
then their exists at least one point x = c in (a, b) such that f
'(c ) = 0
In this theorem, y = f (x) is continuous on [a, b] and differentiable in (a, b). Also f (b) = f (a). Hence the curve y = f (x) can be drawn as in the adjoining figure. The result f ‘ ( c ) = 0 means the tangent at x = c is parallel to x  axis. Hence y = f (x) is continuous on [a,b], differentiable in (a, b) and f (b) = f (a), then there exists at least one point on this curve at which the tangent is parallel to x  axis.
[next page]
