RD Sharma solutions for Mathematics [English] Class 12 chapter 15 - Mean Value Theorems [Latest edition]

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) = [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 ?

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) = 2x² − 5x + 3 on [1, 3] Discuss the applicability of Rolle's theorem for the following function on the indicated intervals ?

f (x) = x^2/3 on [−1, 1] Discuss the applicability of Rolle's theorem for the following function on the indicated intervals ?

\[f\left( x \right) = \begin{cases}- 4x + 5, & 0 \leq x \leq 1 \\ 2x - 3, & 1 < x \leq 2\end{cases}\] 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 − 8x + 12 on [2, 6] ?

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 − 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 − 1) (x − 2) on [−1, 2] ?

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) = x² + 5x + 6 on the interval [−3, −2]  ?

Verify Rolle's theorem for each of the following function on the indicated interval f (x) = cos 2 (x − π/4) on [0, π/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) = e^x sin x on [0, π] ?

Verify Rolle's theorem for the following function on the indicated interval f(x) = e^x cos x on [−π/2, π/2] ?

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) = \[\frac{\sin x}{e^x}\] on 0 ≤ x ≤ 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) = \[{e^{1 - x}}^2\] on [−1, 1] ?

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] ?

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 \[f\left( x \right) = \frac{x}{2} - \sin\frac{\pi x}{6} \text { on }[ - 1, 0]\]?

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) = 4^sin x on [0, π] ?

Verify Rolle's theorem for the following function on the indicated interval f(x) = x² − 5x + 4 on [1, 4] ?

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 Rolle's theorem for the following function on the indicated interval f(x) = sin x − sin 2x on [0, π]?

Using Rolle's theorem, find points on the curve y = 16 − x², x ∈ [−1, 1], where tangent is parallel to x-axis.

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] ?

At what point  on the following curve, is the tangent parallel to x-axis y = 12 (x + 1) (x − 2) on [−1, 2] ?

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?

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) = x² − 1 on [2, 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) = 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 theorem f(x) = x(x −1) 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) = x² − 3x + 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) = 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 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 theorem f(x) = 2x − x² 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) = (x − 1)(x − 2)(x − 3) 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 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 \[f\left( x \right) = x + \frac{1}{x} \text { on }[1, 3]\] ?

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) = 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] ?

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] ?

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 a point on the parabola y = (x − 4)², where the tangent is parallel to the chord joining (4, 0) and (5, 1) ?

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 points on the curve y = x³ − 3x, where the tangent to the curve is parallel to the chord joining (1, −2) and (2, 2) ?

Find a point on the curve y = x³ + 1 where the tangent is parallel to the chord joining (1, 2) and (3, 28) ?

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).

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 ?

State Lagrange's mean value theorem ?

If the value of c prescribed in Rolle's theorem for the function f (x) = 2x (x − 3)ⁿ on the interval \[[0, 2\sqrt{3}] \text { is } \frac{3}{4},\] write the value of n (a positive integer) ?

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] ?

If the polynomial equation \[a_0 x^n + a_{n - 1} x^{n - 1} + a_{n - 2} x^{n - 2} + . . . + a_2 x^2 + a_1 x + a_0 = 0\] n positive integer, has two different real roots α and β, then between α and β, the equation \[n \ a_n x^{n - 1} + \left( n - 1 \right) a_{n - 1} x^{n - 2} + . . . + a_1 = 0 \text { has }\].

If 4a + 2b + c = 0, then the equation 3ax² + 2bx + c = 0 has at least one real root lying in the interval

For the function f (x) = x + \[\frac{1}{x}\] ∈ [1, 3], the value of c for the Lagrange's mean value theorem is 

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) = a^sin x, a > a in

The value of c in Rolle's theorem when
f (x) = 2x³ − 5x² − 4x + 3, x ∈ [1/3, 3] 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 \[f\left( x \right) = \frac{x\left( x + 1 \right)}{e^x}\] defined on [−1, 0] is

The value of c in Lagrange's mean value theorem for the function f (x) = x (x − 2) when x ∈ [1, 2] 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) = e^x sin x in [0, π], then c in Rolle's theorem is