#### Chapters

Chapter 2: Functions

Chapter 3: Binary Operations

Chapter 4: Inverse Trigonometric Functions

Chapter 5: Algebra of Matrices

Chapter 6: Determinants

Chapter 7: Adjoint and Inverse of a Matrix

Chapter 8: Solution of Simultaneous Linear Equations

Chapter 9: Continuity

Chapter 10: Differentiability

Chapter 11: Differentiation

Chapter 12: Higher Order Derivatives

Chapter 13: Derivative as a Rate Measurer

Chapter 14: Differentials, Errors and Approximations

Chapter 15: Mean Value Theorems

Chapter 16: Tangents and Normals

Chapter 17: Increasing and Decreasing Functions

Chapter 18: Maxima and Minima

Chapter 19: Indefinite Integrals

Chapter 20: Definite Integrals

Chapter 21: Areas of Bounded Regions

Chapter 22: Differential Equations

Chapter 23: Algebra of Vectors

Chapter 24: Scalar Or Dot Product

Chapter 25: Vector or Cross Product

Chapter 26: Scalar Triple Product

Chapter 27: Direction Cosines and Direction Ratios

Chapter 28: Straight Line in Space

Chapter 29: The Plane

Chapter 30: Linear programming

Chapter 31: Probability

Chapter 32: Mean and Variance of a Random Variable

Chapter 33: Binomial Distribution

#### RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

## Chapter 15: Mean Value Theorems

#### Chapter 15: Mean Value Theorems solutions [Pages 8 - 9]

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) = 2x^{2} − 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) = x^{2} − 8x + 12 on [2, 6] ?

Verify Rolle's theorem for the following function on the indicated interval f(x) = *x*^{2} − 4*x* + 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) = (x^{2} − 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) = x^{2} + 5x + 6 on the interval [−3, −2] ?

Verify Rolle's theorem for 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 (x^{2} + 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) = x^{2} − 5x + 4 on [1, 4] ?

Verify Rolle's theorem for the following function on the indicated interval f(x) = sin^{4} x + cos^{4} 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 − *x*^{2}, *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* = *x*^{2} 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] ?

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?

It is given that the Rolle's theorem holds for the function f(x) = x^{3} + bx^{2} + cx, x \[\in\] at the point x = \[\frac{4}{3}\] , Find the values of b and c ?

#### Chapter 15: Mean Value Theorems solutions [Pages 17 - 18]

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^{2} − 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) = x^{3} − 2x^{2} − 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) = x^{2} − 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) = 2x^{2} − 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) = x^{2} − 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 − x^{2} 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) = x^{2}^{ }+ 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) = x^{3} − 5x^{2} − 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 curve *y* = *x*^{2} + *x*, where the tangent is parallel to the chord joining (0, 0) and (1, 2) ?

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* = *x*^{3} − 3*x*, where the tangent to the curve is parallel to the chord joining (1, −2) and (2, 2) ?

Find a point on the curve *y* = *x*^{3} + 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) sec^{2} a < tan b − tan a < (b − a) sec^{2} b

where 0 < a < b < \[\frac{\pi}{2}\] ?

#### Chapter 15: Mean Value Theorems solutions [Page 19]

If f (x) = Ax^{2} + Bx + C is such that f (a) = f (b), then write the value of *c* in Rolle's theorem ?

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] ?

#### Chapter 15: Mean Value Theorems solutions [Pages 19 - 20]

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 \[n \ a_n x^{n - 1} + \left( n - 1 \right) a_{n - 1} x^{n - 2} + . . . + a_1 = 0 \text { has }\].

(a) exactly one root

(b) almost one root

(c) at least one root

(d) no root

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

(a) (0, 1)

(b) (1, 2)

(c) (0, 2)

(d) none of these

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

(a) 1

(b) \[\sqrt{3}\]

(c) 2

(d) none of these

If from Lagrange's mean value theorem, we have \[f' \left( x_1 \right) = \frac{f' \left( b \right) - f \left( a \right)}{b - a}, \text { then }\]

(a) a < x_{1} ≤ b

(b) a ≤ x_{1} < b

(c) a < x_{1} < b

(d) a ≤ x_{1} ≤ b

Rolle's theorem is applicable in case of ϕ (*x*) = a^{sin}^{ x}, a > a in

(a) any interval

(b) the interval [0, π]

(c) the interval (0, π/2)

(d) none of these

The value of *c* in Rolle's theorem when

f (x) = 2x^{3} − 5x^{2} − 4x + 3, x ∈ [1/3, 3] is

(a) 2

(b)\[- \frac{1}{3}\]

(c) −2

(d) \[\frac{2}{3}\]

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

(a) *e*^{1}^{/1−e}

(b) *e*^{(e−1)(2e−1)}

(c) \[e^\frac{2e - 1}{e - 1}\]

(d) \[\frac{e - 1}{e}\]

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) 0.5

(b) \[\frac{1 + \sqrt{5}}{2}\]

(c)\[\frac{1 - \sqrt{5}}{2}\]

(d) −0.5

The value of *c* in Lagrange's mean value theorem for the function *f* (*x*) = *x* (*x* − 2) when *x* ∈ [1, 2] is

(a) 1

(b) 1/2

(c) 2/3

(d) 3/2

\[c = \frac{3}{2} \in \left( 1, 2 \right)\]The value of *c* in Rolle's theorem for the function f (x) = x^{3} − 3x in the interval [0,\[\sqrt{3}\]] is

(a) 1

(b) −1

(c) 3/2

(d) 1/3

If f (x) = e^{x} sin x in [0, π], then c in Rolle's theorem is

(a) π/6

(b) π/4

(c) π/2

(d) 3π/4

#### RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

#### Textbook solutions for Class 12

## RD Sharma solutions for Class 12 Mathematics chapter 15 - Mean Value Theorems

RD Sharma solutions for Class 12 Maths chapter 15 (Mean Value Theorems) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 12 Mathematics chapter 15 Mean Value Theorems are Maximum and Minimum Values of a Function in a Closed Interval, Maxima and Minima, Simple Problems on Applications of Derivatives, Graph of Maxima and Minima, Approximations, Tangents and Normals, Increasing and Decreasing Functions, Rate of Change of Bodies Or Quantities, Introduction to Applications of Derivatives.

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