Advertisements
Advertisements
प्रश्न
The value of c in Rolle's theorem when
f (x) = 2x3 − 5x2 − 4x + 3, x ∈ [1/3, 3] is
पर्याय
2
\[- \frac{1}{3}\]
−2
\[\frac{2}{3}\]
Advertisements
उत्तर
2
Given: \[f\left( x \right) = 2 x^3 - 5 x^2 - 4x + 3\] Differentiating the given function with respect to x, we get
\[f'\left( x \right) = 6 x^2 - 10x - 4\]
\[ \Rightarrow f'\left( c \right) = 6 c^2 - 10c - 4\]
\[ \therefore f'\left( c \right) = 0 \]
\[ \Rightarrow 3 c^2 - 5c - 2 = 0\]
\[ \Rightarrow 3 c^2 - 6c + c - 2 = 0\]
\[ \Rightarrow 3c\left( c - 2 \right) + c - 2 = 0\]
\[ \Rightarrow \left( 3c + 1 \right)\left( c - 2 \right) = 0\]
\[ \Rightarrow c = 2, \frac{- 1}{3}\]
\[ \therefore c = 2 \in \left( \frac{1}{3}, 3 \right)\]
Thus,\[c = 2 \in \left( \frac{1}{3}, 3 \right)\] for which Rolle's theorem holds.
Hence, the required value of c is 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)
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 ______.
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) = (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 + 5x + 6 on the interval [−3, −2] ?
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) = 2 sin x + sin 2x on [0, π] ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = sin4 x + cos4 x on \[\left[ 0, \frac{\pi}{2} \right]\] ?
At what point on the following curve, is the tangent parallel to x-axis y = x2 on [−2, 2]
?
At what point on the following curve, is the tangent parallel to x-axis y = \[e^{1 - x^2}\] on [−1, 1] ?
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) = 2x2 − 3x + 1 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 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(x) = x(x + 4)2 on [0, 4] ?
Find a point on the parabola y = (x − 4)2, where the tangent is parallel to the chord joining (4, 0) and (5, 1) ?
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).
State Rolle's theorem ?
State Lagrange's mean value theorem ?
If from Lagrange's mean value theorem, we have \[f' \left( x_1 \right) = \frac{f' \left( b \right) - f \left( a \right)}{b - a}, \text { then }\]
Rolle's theorem is applicable in case of ϕ (x) = asin x, a > a in
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 ?
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?
Show that the local maximum value of `x + 1/x` is less than local minimum value.
An isosceles triangle of vertical angle 2θ is inscribed in a circle of radius a. Show that the area of triangle is maximum when θ = `pi/6`
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`
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 ____________.
The least value of the function f(x) = 2 cos x + x in the closed interval `[0, π/2]` is:
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.
If f(x) = ax2 + 6x + 5 attains its maximum value at x = 1, then the value of a is
