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Sunday, 3 August 2014

CHAPTER 17 - Introduction to Mean Value Theorems

In this section we will deal with some straight forward but quite useful applications of derivatives. We start with the Rolle’s theorem, a simple but powerful theorem having a lot of practical importance.
(A) ROLLE’S THEOREM
Let f(x) be a function defined on [a,b] such that
(i) it is continuous [a,b]
(ii) it is differentiable on (a,b)
(iii) f(a) = f(b)
Then there exists a real number such that f'(c) = 0.
The geometrical interpretation of this theorem is quite straightforward. Consider an arbitrary curve y = f(x) and two points x = a and x = b such that f(a) = f(b).
Since A and B are joined by a continuous and differentiable curve, at least one point x =c will always exist in (a,b) where the tangent drawn is horizontal, or equivalently, f'(c) = 0. Convince yourself that no matter what curve joins Aand B, as long as it is continuous and differentiable one such c will always exist.
From Rolle’s theorem, it follows that between any two roots of a polynomial f(x) will lie a root of the polynomial f'(x).
The (straightforward) proof of Rolle’s theorem is left as an exercise to the reader.
(B) LAGRANGE’S MEAN VALUE THEOREM
Let f(x) be a function defined on [a,b] such that
(i) it is continuous on [a,b]
(ii) it is differentiable on (a,b).
Then there exists a real number c \in \left( {a,b} \right) such that
f'\left( c \right) = \dfrac{{f\left( b \right) - f\left( a \right)}}{{b - a}}
To interpret this theorem geometrically, we take an arbitrary function y = f(x)and two arbitrary points x = a and x = b on it
We see that no matter what the curve between R and P is like, as long as it is continuous and differentiable, there will exist a c \in \left( {a,\,b} \right) such that the tangent drawn at x = c will have a slope equal to \tan \theta  i.e, the average slope from x = a to x = b.
For a rigorous proof of LMVT, consider the function
g\left( x \right) = f\left( x \right) - \left( {\dfrac{{f\left( b \right) - f\left( a \right)}}{{b - a}}} \right)x
Verify that g(x) satisfies all the three criteria of Rolle’s theorem on [a, b] so that
g'\left( c \right) = 0 for at least one c \in \left( {a,\,b} \right)
or
f'\left( c \right) = \dfrac{{f\left( b \right) - f\left( a \right)}}{{b - a}} for at least one c \in \left( {a,b} \right)
Notice that LMVT is an extension of the Rolle’s theorem. In fact, for f(a) = f(b)LMVT reduces to the Rolle’s theorem.
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