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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 $A$and $B$, 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.