We have also introduced the concept of differentiation of such functions and have extended Rolle's theorem and Lagrange's theorem in soft settings.
In this paper we have dealt with continuity and differentiability of functions of soft real sets and extended some celebrated theorems, like Bolzano's theorem, fixed point theorem, intermediate value property, and Rolle's theorem, in soft settings.
In this concrete case, we can extend the results of Theorem 1 to higher derivatives, as no boundary conditions on the same derivative of G(x, t) apply at a and b, which would force a change of sign in the next derivative due to Rolle's theorem
. We will do so in the next result.
It follows from Rolle's theorem
that the derivative x'(t) has a zero, between two zeros of x(t).
Hence, by Rolle's Theorem
, 3c 6 ([z.sub.1],[z.sub.2]) such that [phi]'(c) = 0.
Later chapters cover continuity of functions, derivatives of functions, trigonometric limits, and exponential and logarithmic functions and their derivatives, in addition to implicit functions, parametric functions, Rolle's theorem
, Taylor's formula, and hyperbolic functions.
By Rolle's theorem
, we can choose a < c < [t.sub.1] and [t.sub.1] < d < b such that
Then we use the fact that h satisfies the conditions for Rolle's theorem
to deduce that there is a point c in (a, b) such that h'(c) = 0, and the Mean Value Theorem follows.
Do these possibilities of visual imagining provide a way of discovering Rolle's Theorem
? Here is a statement of the theorem:
We first give a geometric interpretation of how Mean the Value Theorem is proved and simulate the graph to which we normally apply the Rolle's Theorem
. Next, we give a geometric description of how the Cauchy Mean-Value is stated and shed some light on how we can arrive at the function to which Rolle's Theorem
is applied to yield the Cauchy Mean Value Theorem holds.