Advertisements
Advertisements
Question
The value of c in Rolle's theorem for the function f (x) = x3 − 3x in the interval [0,\[\sqrt{3}\]] is
Options
1
−1
3/2
1/3
Solution
1
The given function is \[f\left( x \right) = x^3 - 3x\] .
This is a polynomial function, which is continuous and derivable in R.
Therefore, the function is continuous on [0,\[\sqrt{3}\]] and derivable on (0,\[\sqrt{3}\])
Differentiating the given function with respect to x, we get
\[f'\left( x \right) = 3 x^2 - 3\]
\[ \Rightarrow f'\left( c \right) = 3 c^2 - 3\]
\[ \therefore f'\left( c \right) = 0 \]
\[ \Rightarrow 3 c^2 - 3 = 0\]
\[ \Rightarrow c^2 = 1\]
\[ \Rightarrow c = \pm 1\]
Thus, \[c = 1 \in \left[ 0, \sqrt{3} \right]\] for which Rolle's theorem holds.
Hence, the required value of c is 1.
APPEARS IN
RELATED QUESTIONS
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
Verify Rolle's theorem for the following function on the indicated interval f (x) = x(x − 4)2 on the interval [0, 4] ?
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) = cos 2x 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] ?
At what point on the following curve, is the tangent parallel to x-axis y = \[e^{1 - x^2}\] on [−1, 1] ?
At what point on the following curve, is the tangent parallel to x-axis y = 12 (x + 1) (x − 2) on [−1, 2] ?
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 \[f\left( x \right) = x + \frac{1}{x} \text { on }[1, 3]\] ?
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 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] ?
Find a point on the parabola y = (x − 3)2, where the tangent is parallel to the chord joining (3, 0) and (4, 1) ?
Find a point on the curve y = x3 + 1 where the tangent is parallel to the chord joining (1, 2) and (3, 28) ?
If f (x) = Ax2 + Bx + C is such that f (a) = f (b), then write the value of c in Rolle's theorem ?
State Rolle's theorem ?
State Lagrange's mean value theorem ?
Find the value of c prescribed by Lagrange's mean value theorem for the function \[f\left( x \right) = \sqrt{x^2 - 4}\] defined on [2, 3] ?
The value of c in Rolle's theorem when
f (x) = 2x3 − 5x2 − 4x + 3, x ∈ [1/3, 3] 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
Find the points on the curve x2 + y2 − 2x − 3 = 0 at which the tangents are parallel to the x-axis ?
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`.
Find the maximum and minimum values of f(x) = secx + log cos2x, 0 < x < 2π
Find the area of greatest rectangle that can be inscribed in an ellipse `x^2/"a"^2 + y^2/"b"^2` = 1
The values of a for which y = x2 + ax + 25 touches the axis of x are ______.
The least value of the function f(x) = `"a"x + "b"/x` (where a > 0, b > 0, x > 0) is ______.
Let y = `f(x)` be the equation of a curve. Then the equation of tangent at (xo, yo) is :-
