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) = 2x − x2 on [0, 1] ?
Advertisements
उत्तर
We have,
\[f\left( x \right) = 2x - x^2\]
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
\[f'\left( c \right) = \frac{f\left( 1 \right) - f\left( 0 \right)}{1 - 0} = \frac{f\left( 1 \right) - f\left( 0 \right)}{1}\]
Now, \[f\left( x \right) = 2x - x^2\]
\[\Rightarrow f'\left( x \right) = 2 - 2x\],\[f\left( 1 \right) = 2 - 1 = 1\],\[f\left( 0 \right) = 0\]
∴ \[f'\left( x \right) = \frac{f\left( 1 \right) - f\left( 0 \right)}{1}\]
\[\Rightarrow 2 - 2x = \frac{1 - 0}{1}\]
\[ \Rightarrow - 2x = 1 - 2\]
\[ \Rightarrow x = \frac{1}{2}\]
Thus,
\[c = \frac{1}{2} \in \left( 0, 1 \right)\] 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 local maxima and local minima, of the function f(x) = sin x − cos x, 0 < x < 2π.
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) = 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 ?
f (x) = x2/3 on [−1, 1] 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 − 4x + 3 on [1, 3] ?
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) = ex cos x on [−π/2, π/2] ?
Verify Rolle's theorem for the following function on the indicated interval f (x) = \[\frac{\sin x}{e^x}\] on 0 ≤ x ≤ π ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = sin x + cos x on [0, π/2] ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = sin4 x + cos4 x on \[\left[ 0, \frac{\pi}{2} \right]\] ?
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 − 2x + 4 on [1, 5] ?
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 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) = sin x − sin 2x − x on [0, π] ?
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] ?
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 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] ?
For the function f (x) = x + \[\frac{1}{x}\] ∈ [1, 3], the value of c for the Lagrange's mean value theorem is
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
If f (x) = ex sin x in [0, π], then c in Rolle's theorem 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.
Find the difference between the greatest and least values of the function f(x) = sin2x – x, on `[- pi/2, pi/2]`
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`
Prove that f(x) = sinx + `sqrt(3)` cosx has maximum value at x = `pi/6`
If the graph of a differentiable function y = f (x) meets the lines y = – 1 and y = 1, then the graph ____________.
What does continuity on a closed interval guarantee?
For a continuous function on the closed interval \[a,d\], what does \[f(b)\] represent?
Which statement expresses the Extreme Value Theorem?
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 \[f\left(\frac{1}{8}\right)\] 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 \[-1,1\]?
