Advertisements
Advertisements
प्रश्न
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
उत्तर
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
संबंधित प्रश्न
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.
A cone is inscribed in a sphere of radius 12 cm. If the volume of the cone is maximum, find its height
f(x) = 3 + (x − 2)2/3 on [1, 3] 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 ?
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) = x2 − 4x + 3 on [1, 3] ?
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) = x(x − 1)2 on [0, 1] ?
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) = x(x −2)2 on the interval [0, 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) = 4sin x on [0, π] ?
At what point on the following curve, is the tangent parallel to x-axis y = x2 on [−2, 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) = 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 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(x) = tan−1 x 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 \[f\left( x \right) = \sqrt{x^2 - 4} \text { on }[2, 4]\] ?
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) = x3 − 5x2 − 3x on [1, 3] ?
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) ?
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
The value of c in Lagrange's mean value theorem for the function f (x) = x (x − 2) when x ∈ [1, 2] is
Find the points on the curve x2 + y2 − 2x − 3 = 0 at which the tangents are parallel to the x-axis ?
Find the maximum and minimum values of f(x) = secx + log cos2x, 0 < x < 2π
If f(x) = `1/(4x^2 + 2x + 1)`, then its maximum value is ______.
The maximum value of sinx + cosx 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 ______.
The least value of the function f(x) = `"a"x + "b"/x` (where a > 0, b > 0, x > 0) is ______.
If the graph of a differentiable function y = f (x) meets the lines y = – 1 and y = 1, then the graph ____________.
It is given that at x = 1, the function x4 - 62x2 + ax + 9 attains its maximum value on the interval [0, 2]. Find the value of a.
Let y = `f(x)` be the equation of a curve. Then the equation of tangent at (xo, yo) is :-
After finding critical points and identifying endpoints, what should be done next?
Which list contains all critical points and endpoints for \[f(x)=12x^{\frac{4}{3}}-6x^{\frac{1}{3}}\] on \[-1,1\]?
