Advertisements
Advertisements
प्रश्न
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] ?
Advertisements
उत्तर
We have,
\[f\left( x \right) = x^3 - 2 x^2 - x + 3 = 0\]
Since a polynomial function is everywhere continuous and differentiable.
Therefore,
\[f\left( x \right)\] is continuous on \[\left[ 0, 1 \right]\] and differentiable on \[\left( 0, 1 \right)\]
Thus, both conditions of Lagrange's mean value theorem are satisfied.
So, there must exist at least one real number \[c \in \left( 0, 1 \right)\] such that \[f'\left( c \right) = \frac{f\left( 1 \right) - f\left( 0 \right)}{1 - 0}\]
Now, \[f\left( x \right) = x^3 - 2 x^2 - x + 3 = 0\]
\[\Rightarrow f'\left( x \right) = 3 x^2 - 4x - 1\] ,
\[\Rightarrow 3 x^2 - 4x - 1 = \frac{1 - 3}{1}\]
\[ \Rightarrow 3 x^2 - 4x - 1 + 2 = 0\]
\[ \Rightarrow 3 x^2 - 4x + 1 = 0\]
\[ \Rightarrow 3 x^2 - 3x - x + 1 = 0\]
\[ \Rightarrow \left( 3x - 1 \right)\left( x - 1 \right) = 0\]
\[ \Rightarrow x = \frac{1}{3}, 1\]
Thus,
\[c = \frac{1}{3} \in \left( 0, 1 \right)\] such that \[f'\left( c \right) = \frac{f\left( 1 \right) - f\left( 0 \right)}{1 - 0}\] .Hence, Lagrange's theorem is verified.
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)
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 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) = [x] for −1 ≤ x ≤ 1, where [x] denotes the greatest integer not exceeding x 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 + 5x + 6 on the interval [−3, −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) = 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 \[f\left( x \right) = \frac{x}{2} - \sin\frac{\pi x}{6} \text { on }[ - 1, 0]\]?
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] ?
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] ?
Discuss the applicability of Lagrange's mean value theorem for the function
f(x) = | x | on [−1, 1] ?
Verify the hypothesis and conclusion of Lagrange's man value theorem for the function
f(x) = \[\frac{1}{4x - 1},\] 1≤ x ≤ 4 ?
State Rolle's theorem ?
If 4a + 2b + c = 0, then the equation 3ax2 + 2bx + c = 0 has at least one real root lying in the interval
If from Lagrange's mean value theorem, we have \[f' \left( x_1 \right) = \frac{f' \left( b \right) - f \left( a \right)}{b - a}, \text { then }\]
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
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 ______.
The maximum value of sinx + cosx is ______.
Prove that f(x) = sinx + `sqrt(3)` cosx has maximum value at x = `pi/6`
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
For a continuous function on the closed interval \[a,d\], what does \[f(c)\] represent?
Which statement about local extrema and absolute extrema is correct?
If a differentiable function has an absolute max or min at an interior point \[c\], what must be true?
Which points must always be checked when finding absolute extrema on a closed interval \[a,b\]?
How are the absolute maximum and absolute minimum determined after evaluating \[f(x)\] at all critical points and endpoints?
For \[f(x)=12x^{\frac{4}{3}}-6x^{\frac{1}{3}}\], which derivative is correct?
What is \[f(-1)\] for \[f(x)=12x^{\frac{4}{3}}-6x^{\frac{1}{3}}\]?
What is \[f\left(\frac{1}{8}\right)\] for \[f(x)=12x^{\frac{4}{3}}-6x^{\frac{1}{3}}\]?
What is \[f(0)\] for \[f(x)=12x^{\frac{4}{3}}-6x^{\frac{1}{3}}\]?
What are the absolute extrema of \[f(x)=12x^{\frac{4}{3}}-6x^{\frac{1}{3}}\] on \[-1,1\]?
