Advertisements
Advertisements
Question
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] ?
Solution
We have,
Therefore, \[f\left( x \right)\] is continuous on \[\left[ 1, 3 \right]\] and differentiable on \[\left( 1, 3 \right)\].
So, there must exist at least one real number \[c \in \left( 1, 3 \right)\] such that\[f'\left( c \right) = \frac{f\left( 3 \right) - f\left( 1 \right)}{3 - 1} = \frac{f\left( 3 \right) - f\left( 1 \right)}{2}\]
\[ \Rightarrow 4x - 3 - 5 = 0\]
\[ \Rightarrow x = 2\]
Thus,
\[c = 2 \in \left( 1, 3 \right)\] such that \[f'\left( c \right) = \frac{f\left( 3 \right) - f\left( 1 \right)}{3 - 1}\]
APPEARS IN
RELATED QUESTIONS
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 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) = sin \[\frac{1}{x}\] for −1 ≤ x ≤ 1 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) = x(x − 4)2 on the interval [0, 4] ?
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) = ex cos x on [−π/2, π/2] ?
Verify Rolle's theorem for the following function on the indicated interval f (x) = \[{e^{1 - x}}^2\] on [−1, 1] ?
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 \[f\left( x \right) = \frac{6x}{\pi} - 4 \sin^2 x \text { on } [0, \pi/6]\] ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = sin x − sin 2x on [0, π]?
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) ?
Examine if Rolle's theorem is applicable to any one of the following functions.
(i) f (x) = [x] for x ∈ [5, 9]
(ii) f (x) = [x] for x ∈ [−2, 2]
Can you say something about the converse of Rolle's Theorem from these functions?
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] ?
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] ?
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) = x2 + x − 1 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(x) = sin x − sin 2x − x on [0, π] ?
Show that the lagrange's mean value theorem is not applicable to the function
f(x) = \[\frac{1}{x}\] 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 ?
Find the points on the curve y = x3 − 3x, where the tangent to the curve is parallel to the chord joining (1, −2) and (2, 2) ?
Using Lagrange's mean value theorem, prove that (b − a) sec2 a < tan b − tan a < (b − a) sec2 b
where 0 < a < b < \[\frac{\pi}{2}\] ?
State Rolle's theorem ?
Rolle's theorem is applicable in case of ϕ (x) = asin x, a > a in
Show that the local maximum value of `x + 1/x` is less than local minimum value.
Find the difference between the greatest and least values of the function f(x) = sin2x – x, on `[- pi/2, pi/2]`
Minimum value of f if f(x) = sinx in `[(-pi)/2, pi/2]` is ______.
The maximum value of sinx + cosx is ______.
Prove that f(x) = sinx + `sqrt(3)` cosx has maximum value at x = `pi/6`
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 :-
