Advertisements
Advertisements
Question
f (x) = sin \[\frac{1}{x}\] for −1 ≤ x ≤ 1 Discuss the applicability of Rolle's theorem for the following function on the indicated intervals ?
Advertisements
Solution
The given function is \[f\left( x \right) = \sin\frac{1}{x}\] .
The domain of f is given to be \[\left[ - 1, 1 \right]\] .
It is known that \[\lim_{x \to 0} \sin\frac{1}{x}\] does not exist.
Thus, \[f\left( x \right)\] is discontinuous at x = 0 on \[\left[ - 1, 1 \right]\] .
Hence, Rolle's theorem is not applicable for the given function.
APPEARS IN
RELATED QUESTIONS
Find the local maxima and local minima, of the function f(x) = sin x − cos x, 0 < x < 2π.
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 cylinder is `4/27 pih^3` tan2α.
f (x) = x2/3 on [−1, 1] 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 − 1) (x − 2)2 on [1, 2] ?
Verify Rolle's theorem for the following function on the indicated interval f (x) = (x2 − 1) (x − 2) on [−1, 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 each of the following function on the indicated interval f (x) = cos 2 (x − π/4) on [0, π/2] ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = ex sin x on [0, π] ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = cos 2x on [0, π] ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = sin 3x on [0, π] ?
Verify Rolle's theorem for the following function on the indicated interval f (x) = log (x2 + 2) − log 3 on [−1, 1] ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = sin x + cos x on [0, π/2] ?
Using Rolle's theorem, find points on the curve y = 16 − x2, x ∈ [−1, 1], where tangent is parallel to x-axis.
If f : [−5, 5] → R is differentiable and if f' (x) doesnot vanish anywhere, then prove that f (−5) ± f (5) ?
It is given that the Rolle's theorem holds for the function f(x) = x3 + bx2 + cx, x \[\in\] at the point x = \[\frac{4}{3}\] , Find the values of b and c ?
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 − 1 on [2, 3] ?
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) = x2 + x − 1 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(x) = sin x − sin 2x − x on [0, π] ?
Discuss the applicability of Lagrange's mean value theorem for the function
f(x) = | x | on [−1, 1] ?
Find a point on the parabola y = (x − 4)2, where the tangent is parallel to the chord joining (4, 0) and (5, 1) ?
Find a point on the curve y = x2 + x, where the tangent is parallel to the chord joining (0, 0) and (1, 2) ?
Find a point on the parabola y = (x − 3)2, where the tangent is parallel to the chord joining (3, 0) and (4, 1) ?
State Lagrange's mean value theorem ?
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 4a + 2b + c = 0, then the equation 3ax2 + 2bx + c = 0 has at least one real root lying in the interval
Rolle's theorem is applicable in case of ϕ (x) = asin x, a > a in
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 (x) = x3 − 3x in the interval [0,\[\sqrt{3}\]] is
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.
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 ______.
If the graph of a differentiable function y = f (x) meets the lines y = – 1 and y = 1, then the graph ____________.
If f(x) = ax2 + 6x + 5 attains its maximum value at x = 1, then the value of a is
The minimum value of `1/x log x` in the interval `[2, oo]` is
