Advertisements
Advertisements
Question
Examine if Rolle's theorem is applicable to any one of the following functions.
(i) f (x) = [x] for x ∈ [5, 9]
(ii) f (x) = [x] for x ∈ [−2, 2]
Can you say something about the converse of Rolle's Theorem from these functions?
Advertisements
Solution
By Rolle’s theorem, for a function f : [a , b] →R , if
(a) f is continuous on [a, b],
(b) f is differentiable on (a, b) and
(c) f (a) = f (b),
then there exists some c ∈ (a, b) such that f'(c) = 0 .
Therefore, Rolle’s theorem is not applicable to those functions that do not satisfy any of the three conditions of the hypothesis
(i) f(x) = [x] for x ∈ [ 5 , 9 ]
It is evident that the given function f (x) is not continuous at every integral point.
In particular, f(x) is not continuous at x = 5 and x = 9.
Thus, f (x) is not continuous on [5, 9].
Also , f (5) = [5] = 5 and f (9) = [9] = 9
∴ f (5) ≠ f (9)
The differentiability of f on (5, 9) is checked in the following way.
Let n be an integer such that n ∈ (5, 9).
The left hand limit of f at x = n is,
`lim_(h ->o)(f (n + h) - f (n)\)/h = lim_(h->o) ([n + h ] - [n])/h) = lim_(h->o)(n- 1- n)/h = lim_(h ->o)(-1)/h =oo`
The right hand limit of f at x = n is,
`lim_(h->o) (f (n +h )- f (n))/h = lim _(h->o)([n +h] - [n])/h = lim_(h->o)(n-n)/h = lim _(h->o) 0 = 0 `
Since the left and the right hand limits of f at x = n are not equal, f is not differentiable at x = n.
Thus, f is not differentiable on (5, 9).
It is observed that f does not satisfy all the conditions of the hypothesis of Rolle’s theorem.
Hence, Rolle’s theorem is not applicable on f (x) for x ∈ [5 , 9] .
(ii) f (x) = [x] for x ∈ [-2 , 2]
It is evident that the given function f (x) is not continuous at every integral point.
In particular, f(x) is not continuous at x = −2 and x = 2.
Thus, f (x) is not continuous on [−2, 2].
Also, f (-2) = [-2] = -2 and f (2) = [2] = 2
∴ f (-2) ≠ f (2)
The differentiability of f on (−2, 2) is checked in the following way.
Let n be an integer such that n ∈ (−2, 2).
The left hand limit of f at x = n is .
`lim_(h->o) (f (n + h) -f (n))/ h = lim_(h->o) ([n +h]- [n])/h = lim_(h->o)(n - 1- n)/h = lim_(h->o) (-1)/ h =oo`
The right hand limit of f at x = n is ,
`lim_(h->o) (f(n+h) -f (n))/h = lim_(h->o)([n+h] - [n])/h) lim_(h->o)(n-n)/h = lim_(h->o)(-1)/h = oo`
Since the left and the right hand limits of f at x = n are not equal, f is not differentiable at x = n.
Thus, f is not differentiable on (−2, 2).
It is observed that f does not satisfy all the conditions of the hypothesis of Rolle’s theorem.
Hence, Rolle’s theorem is not applicable on f(x)= [x] for x ∈ [ -2 , 2 ].
APPEARS IN
RELATED QUESTIONS
f (x) = [x] for −1 ≤ x ≤ 1, where [x] denotes the greatest integer not exceeding x Discuss the applicability of Rolle's theorem for the following function on the indicated intervals ?
f (x) = 2x2 − 5x + 3 on [1, 3] Discuss the applicability of Rolle's theorem for the following function on the indicated intervals ?
\[f\left( x \right) = \begin{cases}- 4x + 5, & 0 \leq x \leq 1 \\ 2x - 3, & 1 < x \leq 2\end{cases}\] Discuss the applicability of Rolle's theorem for the following function on the indicated intervals ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = x(x −2)2 on the interval [0, 2] ?
Verify Rolle's theorem for the following function on the indicated interval f (x) = x2 + 5x + 6 on the interval [−3, −2] ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = sin 2x on [0, π/2] ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = cos 2x on [−π/4, π/4] ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = ex cos x on [−π/2, π/2] ?
Verify Rolle's theorem for the following function on the indicated interval \[f\left( x \right) = \frac{x}{2} - \sin\frac{\pi x}{6} \text { on }[ - 1, 0]\]?
Verify Rolle's theorem for the following function on the indicated interval \[f\left( x \right) = \frac{6x}{\pi} - 4 \sin^2 x \text { on } [0, \pi/6]\] ?
Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theore f(x) = (x − 1)(x − 2)(x − 3) on [0, 4] ?
Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theore \[f\left( x \right) = \sqrt{25 - x^2}\] on [−3, 4] ?
Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem f(x) = sin x − sin 2x − x on [0, π] ?
Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem f(x) = x3 − 5x2 − 3x on [1, 3] ?
Discuss the applicability of Lagrange's mean value theorem for the function
f(x) = | x | on [−1, 1] ?
Show that the lagrange's mean value theorem is not applicable to the function
f(x) = \[\frac{1}{x}\] on [−1, 1] ?
Find the points on the curve y = x3 − 3x, where the tangent to the curve is parallel to the chord joining (1, −2) and (2, 2) ?
Let C be a curve defined parametrically as \[x = a \cos^3 \theta, y = a \sin^3 \theta, 0 \leq \theta \leq \frac{\pi}{2}\] . Determine a point P on C, where the tangent to C is parallel to the chord joining the points (a, 0) and (0, a).
If the value of c prescribed in Rolle's theorem for the function f (x) = 2x (x − 3)n on the interval \[[0, 2\sqrt{3}] \text { is } \frac{3}{4},\] write the value of n (a positive integer) ?
If the polynomial equation \[a_0 x^n + a_{n - 1} x^{n - 1} + a_{n - 2} x^{n - 2} + . . . + a_2 x^2 + a_1 x + a_0 = 0\] n positive integer, has two different real roots α and β, then between α and β, the equation \[n \ a_n x^{n - 1} + \left( n - 1 \right) a_{n - 1} x^{n - 2} + . . . + a_1 = 0 \text { has }\].
For the function f (x) = x + \[\frac{1}{x}\] ∈ [1, 3], the value of c for the Lagrange's mean value theorem is
When the tangent to the curve y = x log x is parallel to the chord joining the points (1, 0) and (e, e), the value of x is ______.
The value of c in Rolle's theorem for the function \[f\left( x \right) = \frac{x\left( x + 1 \right)}{e^x}\] defined on [−1, 0] is
Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height h and semi-vertical angle α is one-third that of the cone and the greatest volume of the cylinder is `(4)/(27) pi"h"^3 tan^2 α`.
The values of a for which y = x2 + ax + 25 touches the axis of x are ______.
If f(x) = `1/(4x^2 + 2x + 1)`, then its maximum value is ______.
Prove that f(x) = sinx + `sqrt(3)` cosx has maximum value at x = `pi/6`
At x = `(5pi)/6`, f(x) = 2 sin3x + 3 cos3x is ______.
If the graph of a differentiable function y = f (x) meets the lines y = – 1 and y = 1, then the graph ____________.
The minimum value of `1/x log x` in the interval `[2, oo]` is
What does continuity on a closed interval guarantee?
For a continuous function on the closed interval \[a,d\], what does \[f(a)\] represent?
For a continuous function on the closed interval \[a,d\], what does \[f(c)\] represent?
Which points are critical points for a function \[f\]?
After finding critical points and identifying endpoints, what should be done next?
How are the absolute maximum and absolute minimum determined after evaluating \[f(x)\] at all critical points and endpoints?
Why is \[x=0\] a critical point for \[f(x)=12x^{\frac{4}{3}}-6x^{\frac{1}{3}}\]?
