Name: Derivative of Functions in Parametric Form
Uploaded: 2019-07-19
Duration: 8 min 37 s
Description: Learn everything you'll need to know about derivatives of functions in parametric forms with this quick and easy lesson that'll cover the concept comprehensi...

Question

State Rolle's theorem and give geometrical interpretation of the theorem.

Accepted Answer

A special case of Lagrange's mean value theorem is Rolle's Theorem which states that:

If a function $f$ is defined in the closed interval $[a, b]$ in such a way that it satisfies the following conditions.

i) The function $f$ is continuous on the closed interval $[a, b]$

ii)The function $f$ is differentiable on the open interval $(a, b)$

iii) Now if $f (a) = f (b)$ , then there exists at least one value of $x$ , let us assume this value to be $c$ , which lies between $a$ and $b$ i.e. $(a < c < b)$ in such a way that $f (c) = 0$ .

Precisely, if a function is continuous on the closed interval $[a, b]$ and differentiable on the open interval $(a, b)$ then there exists a point $x = c$ in $(a, b)$ such that $f^{'} (c) = 0$ .

Geometric interpretation of Rolle's theorem.

In the given graph, the curve $y = f (x)$ is continuous between $x = a$ and $x = b$ and at every point, within the interval, it is possible to draw a tangent and ordinates corresponding to the abscissa and are equal then there exists at least one tangent to the curve which is parallel to the $x -$ axis.

Algebraically, this theorem tells us that if $f (x)$ is representing a polynomial function in $x$ and the two roots of the equation $f (x) = 0$ are $x = a$ and $x = b$ , then there exists at least one root of the equation $f (x) = 0$ lying between these values.

The converse of Rolle's theorem is not true and it is also possible that there exists more than one value of $x$ , for which the theorem holds good but there is a definite chance of the existence of one such value.

State Rolle's theorem and give geometrical interpretation of the theorem.

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