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

## Chapter 15: Mean Value Theorems

Ex. 15.1Ex. 15.2Others

#### Chapter 15: Mean Value Theorems Exercise 15.1, 15.2 solutions [Pages 8 - 9]

Ex. 15.1 | Q 1.1 | Page 8

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 ?

Ex. 15.1 | Q 1.2 | Page 8

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 ?

Ex. 15.1 | Q 1.3 | Page 8

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 ?

Ex. 15.1 | Q 1.4 | Page 8

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

Ex. 15.1 | Q 1.5 | Page 8

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

Ex. 15.1 | Q 1.6 | Page 8

$f\left( x \right) = \begin{cases}- 4x + 5, & 0 \leq x \leq 1 \\ 2x - 3, & 1 < x \leq 2\end{cases}$ Discuss the applicability of Rolle's theorem for the following function on the indicated intervals ?

Ex. 15.1 | Q 2.1 | Page 9

Verify Rolle's theorem for the following function on the indicated interval f(x) = x2 − 8x + 12 on [2, 6] ?

Ex. 15.1 | Q 2.2 | Page 9

Verify Rolle's theorem for the following function on the indicated interval f(x) = x2 − 4x + 3 on [1, 3] ?

Ex. 15.1 | Q 2.3 | Page 9

Verify Rolle's theorem for the following function on the indicated interval f (x) = (x − 1) (x − 2)2 on [1, 2] ?

Ex. 15.1 | Q 2.4 | Page 9

Verify Rolle's theorem for the following function on the indicated interval  f (x) = x(x − 1)2 on [0, 1] ?

Ex. 15.1 | Q 2.5 | Page 9

Verify Rolle's theorem for the following function on the indicated interval  f (x) = (x2 − 1) (x − 2) on [−1, 2] ?

Ex. 15.1 | Q 2.6 | Page 9

Verify Rolle's theorem for the following function on the indicated interval   f (x) = x(x − 4)2 on the interval [0, 4] ?

Ex. 15.1 | Q 2.7 | Page 9

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

Ex. 15.1 | Q 2.8 | Page 9

Verify Rolle's theorem for the following function on the indicated interval f (x) = x2 + 5x + 6 on the interval [−3, −2]  ?

Ex. 15.1 | Q 3.01 | Page 9

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] ?

Ex. 15.1 | Q 3.02 | Page 9

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

Ex. 15.1 | Q 3.03 | Page 9

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

Ex. 15.1 | Q 3.04 | Page 9

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

Ex. 15.1 | Q 3.05 | Page 9

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

Ex. 15.1 | Q 3.06 | Page 9

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

Ex. 15.1 | Q 3.07 | Page 9

Verify Rolle's theorem for the following function on the indicated interval f (x) = $\frac{\sin x}{e^x}$ on 0 ≤ x ≤ π ?

Ex. 15.1 | Q 3.08 | Page 9

Verify Rolle's theorem for the following function on the indicated interval f(x) = sin 3x on [0, π] ?

Ex. 15.1 | Q 3.09 | Page 9

Verify Rolle's theorem for the following function on the indicated interval f (x) = ${e^{1 - x}}^2$ on [−1, 1] ?

Ex. 15.1 | Q 3.1 | Page 9

Verify Rolle's theorem for the following function on the indicated interval f (x) = log (x2 + 2) − log 3 on [−1, 1] ?

Ex. 15.1 | Q 3.11 | Page 9

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

Ex. 15.1 | Q 3.12 | Page 9

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

Ex. 15.1 | Q 3.13 | Page 9

Verify Rolle's theorem for the following function on the indicated interval $f\left( x \right) = \frac{x}{2} - \sin\frac{\pi x}{6} \text { on }[ - 1, 0]$?

Ex. 15.1 | Q 3.14 | Page 9

Verify Rolle's theorem for the following function on the indicated interval $f\left( x \right) = \frac{6x}{\pi} - 4 \sin^2 x \text { on } [0, \pi/6]$ ?

Ex. 15.1 | Q 3.15 | Page 9

Verify Rolle's theorem for the following function on the indicated interval f(x) = 4sin x on [0, π] ?

Ex. 15.1 | Q 3.16 | Page 9

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

Ex. 15.1 | Q 3.17 | Page 9

Verify Rolle's theorem for the following function on the indicated interval f(x) = sin4 x + cos4 x on $\left[ 0, \frac{\pi}{2} \right]$ ?

Ex. 15.1 | Q 3.18 | Page 9

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

Ex. 15.1 | Q 7 | Page 9

Using Rolle's theorem, find points on the curve y = 16 − x2x ∈ [−1, 1], where tangent is parallel to x-axis.

Ex. 15.2 | Q 8.1 | Page 9

At what point  on the following curve, is the tangent parallel to x-axis y = x2 on [−2, 2]
?

Ex. 15.1 | Q 8.2 | Page 9

At what point  on the following curve, is the tangent parallel to x-axis y = $e^{1 - x^2}$ on [−1, 1] ?

Ex. 15.1 | Q 8.3 | Page 9

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

Ex. 15.1 | Q 9 | Page 9

If f : [−5, 5] → is differentiable and if f' (x) doesnot vanish anywhere, then prove that f (−5) ± f (5) ?

Ex. 15.1 | Q 10 | Page 9

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?

Ex. 15.1 | Q 11 | Page 9

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 ?

#### Chapter 15: Mean Value Theorems Exercise 15.2, 15.1 solutions [Pages 17 - 18]

Ex. 15.2 | Q 1.01 | Page 17

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] ?

Ex. 15.2 | Q 1.02 | Page 17

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] ?

Ex. 15.1 | Q 1.03 | Page 17

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] ?

Ex. 15.2 | Q 1.04 | Page 17

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] ?

Ex. 15.2 | Q 1.05 | Page 17

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] ?

Ex. 15.2 | Q 1.06 | Page 17

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] ?

Ex. 15.2 | Q 1.07 | Page 17

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] ?

Ex. 15.2 | Q 1.08 | Page 17

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] ?

Ex. 15.1 | Q 1.09 | Page 17

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] ?

Ex. 15.2 | Q 1.1 | Page 17

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) = tan1 x on [0, 1] ?

Ex. 15.2 | Q 1.11 | Page 17

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

Ex. 15.2 | Q 1.12 | Page 17

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] ?

Ex. 15.2 | Q 1.13 | Page 17

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 $f\left( x \right) = \sqrt{x^2 - 4} \text { on }[2, 4]$ ?

Ex. 15.2 | Q 1.14 | Page 17

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] ?

Ex. 15.2 | Q 1.15 | Page 17

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] ?

Ex. 15.2 | Q 1.16 | Page 17

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] ?

Ex. 15.2 | Q 2 | Page 18

Discuss the applicability of Lagrange's mean value theorem for the function
f(x) = | x | on [−1, 1] ?

Ex. 15.2 | Q 3 | Page 18

Show that the lagrange's mean value theorem is not applicable to the function
f(x) = $\frac{1}{x}$ on [−1, 1] ?

Ex. 15.2 | Q 4 | Page 18

Verify the  hypothesis and conclusion of Lagrange's man value theorem for the function
f(x) = $\frac{1}{4x - 1},$ 1≤ x ≤ 4 ?

Ex. 15.2 | Q 5 | Page 18

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

Ex. 15.2 | Q 6 | Page 18

Find a point on the curve y = x2 + x, where the tangent is parallel to the chord joining (0, 0) and (1, 2) ?

Ex. 15.2 | Q 7 | Page 18

Find a point on the parabola y = (x − 3)2, where the tangent is parallel to the chord joining (3, 0) and (4, 1) ?

Ex. 15.2 | Q 8 | Page 18

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) ?

Ex. 15.2 | Q 9 | Page 18

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

Ex. 15.2 | Q 10 | Page 18

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).

Ex. 15.2 | Q 11 | Page 18

Using Lagrange's mean value theorem, prove that (b − a) sec2 a < tan b − tan a < (b − a) sec2 b
where 0 < a < b < $\frac{\pi}{2}$ ?

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

Q 1 | Page 19

If f (x) = Ax2 + Bx + C is such that f (a) = f (b), then write the value of c in Rolle's theorem ?

Q 2 | Page 19

State Rolle's theorem ?

Q 3 | Page 19

State Lagrange's mean value theorem ?

Q 4 | Page 19

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) ?

Q 5 | Page 19

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 Exercise 15.2 solutions [Pages 19 - 20]

Q 1 | Page 19

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

• exactly one root

• almost one root

• at least one root

• no root

Q 2 | Page 19

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

• (0, 1)

• (1, 2)

• (0, 2)

• none of these

Q 3 | Page 19

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

• 1

• $\sqrt{3}$

• 2

• none of these

Q 4 | Page 19

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 < x1 ≤ b

•  a ≤ x1 < b

• a < x1 < b

• a ≤ x1 ≤ b

Q 5 | Page 19

Rolle's theorem is applicable in case of ϕ (x) = asin x, a > a in

• any interval

• the interval [0, π]

• the interval (0, π/2)

• none of these

Ex. 15.2 | Q 6 | Page 20

The value of c in Rolle's theorem when
f (x) = 2x3 − 5x2 − 4x + 3, x ∈ [1/3, 3] is

• 2

• $- \frac{1}{3}$

• −2

• $\frac{2}{3}$

Q 7 | Page 20

When the tangent to the curve y = x log x is parallel to the chord joining the points (1, 0) and (ee), the value of x is

• e1/1−e

•  e(e−1)(2e−1)

• $e^\frac{2e - 1}{e - 1}$

• $\frac{e - 1}{e}$

Q 8 | Page 20

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

• 0.5

• $\frac{1 + \sqrt{5}}{2}$

• $\frac{1 - \sqrt{5}}{2}$

• −0.5

Q 9 | Page 20

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

• 1

• 1/2

• 2/3

• 3/2

Q 10 | Page 20

The value of c in Rolle's theorem for the function f (x) = x3 − 3x in the interval [0,$\sqrt{3}$] is

• 1

• −1

• 3/2

• 1/3

Q 11 | Page 20

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

• π/6

• π/4

• π/2

• 3π/4

## Chapter 15: Mean Value Theorems

Ex. 15.1Ex. 15.2Others

## 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.

Further, we at shaalaa.com are providing such solutions so that students can prepare for written exams. RD Sharma textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

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.

Using RD Sharma Class 12 solutions Mean Value Theorems exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in RD Sharma Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 12 prefer RD Sharma Textbook Solutions to score more in exam.

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