Advertisements
Advertisements
प्रश्न
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
उत्तर
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
संबंधित प्रश्न
Find the absolute maximum and absolute minimum values of the function f given by f(x)=sin2x-cosx,x ∈ (0,π)
Find the local maxima and local minima, of the function f(x) = sin x − cos x, 0 < x < 2π.
A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic metre per hour. Then the depth of the wheat is increasing at the rate of ______.
A cone is inscribed in a sphere of radius 12 cm. If the volume of the cone is maximum, find its height
f (x) = 2x2 − 5x + 3 on [1, 3] 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) = x2 − 8x + 12 on [2, 6] ?
Verify Rolle's theorem for the following function on the indicated interval f (x) = x(x − 1)2 on [0, 1] ?
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) = 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(x) = 4sin x on [0, π] ?
At what point on the following curve, is the tangent parallel to x-axis y = \[e^{1 - x^2}\] on [−1, 1] ?
If f : [−5, 5] → R is differentiable and if f' (x) doesnot vanish anywhere, then prove that f (−5) ± f (5) ?
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 − 2x2 − x + 3 on [0, 1] ?
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) = x2 − 3x + 2 on [−1, 2] ?
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) = 2x − x2 on [0, 1] ?
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) = x(x + 4)2 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 theorem \[f\left( x \right) = \sqrt{x^2 - 4} \text { on }[2, 4]\] ?
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] ?
Using Lagrange's mean value theorem, prove that (b − a) sec2 a < tan b − tan a < (b − a) sec2 b
where 0 < a < b < \[\frac{\pi}{2}\] ?
State Lagrange's mean value theorem ?
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
If from Lagrange's mean value theorem, we have \[f' \left( x_1 \right) = \frac{f' \left( b \right) - f \left( a \right)}{b - a}, \text { then }\]
A wire of length 50 m is cut into two pieces. One piece of the wire is bent in the shape of a square and the other in the shape of a circle. What should be the length of each piece so that the combined area of the two is minimum?
A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of types A require 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type B require 8 minutes each for cutting and 4 hours available for assembling. The profit is ₹ 50 each for type A and ₹60 each for type B souvenirs. How many souvenirs of each type should the company manufacture in order to maximize profit? Formulate the above LPP and solve it graphically and find the maximum profit.
Find the difference between the greatest and least values of the function f(x) = sin2x – x, on `[- pi/2, pi/2]`
The values of a for which y = x2 + ax + 25 touches the axis of x are ______.
At x = `(5pi)/6`, f(x) = 2 sin3x + 3 cos3x is ______.
The least value of the function f(x) = `"a"x + "b"/x` (where a > 0, b > 0, x > 0) is ______.
