RD Sharma books are widely used by students who wish to excel in board exams as it provides a vast number of questions to solve. For those students here we have provided the answers for RD Sharma Solutions in simple PDF format, which can be downloaded easily from the below provided links. Solving these questions makes them perfect in all topics included in the syllabus. Knowing the concept is necessary but learning their applications also plays an important role to score full marks in Maths.

The 15th Chapter Mean Value Theorems of RD Sharma Solutions for Class 12 Maths is the best material. This chapter explains the geometrical and algebraic interpretation of some important named theorems related to mean values. Based on these theorems BYJU’S expert faculty has solved this chapter in a unique way. Some of the essential topics of the chapter are listed below.

- Rolle’s theorem
- Geometrical interpretation of Rolle’s theorem
- Algebraic interpretation of Rolle’s theorem
- Checking the applicability of Rolle’s theorem
- Verification of Rolle’s theorem for a given function defined on a given interval
- Miscellaneous applications of Rolle’s theorem
- Lagrange’s mean value theorem
- Geometrical interpretation of Lagrange’s mean value theorem
- Verification of Lagrange’s mean value theorem
- Proving inequalities by using Lagrange’s mean value theorem
- Miscellaneous applications of Lagrange’s mean value theorem

**RD Sharma Solutions Class 12 Maths Chapter 15 Mean Value Theorems:**Download PDF

### Access answers to Maths RD Sharma Solutions For Class 12 Chapter 15 – Mean Value Theorems

Exercise 15.1 Page No: 15.9

**1. Discuss the applicability of Rolleâ€™s Theorem for the following functions on the indicated intervals:**

**Solution:**

**(ii) f (x) = [x] for -1 < x â‰¤ 1, where [x] denotes the greatest integer not exceeding x**

**Solution:**

**(iii) f(x) = sin 1/x for -1 â‰¤ x â‰¤ 1**

**Solution:**

**(iv) f (x) = 2x ^{2} â€“ 5x + 3 on [1, 3]**

**Solution:**

Given function is f (x) = 2x^{2}Â â€“ 5x + 3 on [1, 3]

Since given function f is a polynomial. So, it is continuous and differentiable everywhere.

Now, we find the values of function at the extreme values.

â‡’Â f (1) = 2(1)^{2}â€“5(1) + 3

â‡’Â f (1) = 2 â€“ 5 + 3

â‡’Â f (1) = 0…… (1)

â‡’Â f (3) = 2(3)^{2}â€“5(3) + 3

â‡’Â f (3) = 2(9)â€“15 + 3

â‡’Â f (3) = 18 â€“ 12

â‡’Â f (3) = 6…… (2)

From (1) and (2), we can say that, f (1) â‰ f (3)

âˆ´Â Rolleâ€™s Theorem is not applicable for the function f in interval [1, 3].

**(v) f (x) = x ^{2/3} on [-1, 1]**

**Solution:**

**Solution:**

**2. Verify the Rolleâ€™s Theorem for each of the following functions on the indicated intervals:**

**(i) f (x) = x ^{2} â€“ 8x + 12 on [2, 6]**

**Solution:**

Given function is f (x) = x^{2}Â â€“ 8x + 12 on [2, 6]

Since, given function f is a polynomial it is continuous and differentiable everywhere i.e., on R.

Let us find the values at extremes:

â‡’Â f (2) = 2^{2}Â â€“ 8(2) + 12

â‡’Â f (2) = 4 â€“ 16 + 12

â‡’Â f (2) = 0

â‡’Â f (6) = 6^{2}Â â€“ 8(6) + 12

â‡’Â f (6) = 36 â€“ 48 + 12

â‡’Â f (6) = 0

âˆ´Â f (2) = f(6), Rolleâ€™s theorem applicable for function f on [2,6].

Now we have to find the derivative of f(x)

**(ii) f(x) = x ^{2} â€“ 4x + 3 on [1, 3]**

**Solution:**

Given function is f (x) = x^{2}Â â€“ 4x + 3 on [1, 3]

Since, given function f is a polynomial it is continuous and differentiable everywhere i.e., on R. Let us find the values at extremes:

â‡’Â f (1) = 1^{2}Â â€“ 4(1) + 3

â‡’Â f (1) = 1 â€“ 4 + 3

â‡’Â f (1) = 0

â‡’Â f (3) = 3^{2}Â â€“ 4(3) + 3

â‡’Â f (3) = 9 â€“ 12 + 3

â‡’Â f (3) = 0

âˆ´Â f (1) = f(3), Rolleâ€™s theorem applicable for function â€˜fâ€™ on [1,3].

Letâ€™s find the derivative of f(x)

â‡’Â fâ€™(x) = 2x â€“ 4

We have fâ€™(c) = 0, c Ïµ (1, 3), from the definition of Rolleâ€™s Theorem.

â‡’Â fâ€™(c) = 0

â‡’Â 2c â€“ 4 = 0

â‡’Â 2c = 4

â‡’Â c = 4/2

â‡’Â C = 2 Ïµ (1, 3)

âˆ´Â Rolleâ€™s Theorem is verified.

**(iii) f (x) = (x â€“ 1) (x â€“ 2) ^{2} on [1, 2]**

**Solution:**

Given function is f (x) = (x â€“ 1) (x â€“ 2)^{2}Â on [1, 2]

Since, given function f is a polynomial it is continuous and differentiable everywhere that is on R.

Let us find the values at extremes:

â‡’Â f (1) = (1 â€“ 1) (1 â€“ 2)^{2}

â‡’Â f (1) = 0(1)^{2}

â‡’Â f (1) = 0

â‡’Â f (2) = (2 â€“ 1)(2 â€“ 2)^{2}

â‡’Â f (2) = 0^{2}

â‡’Â f (2) = 0

âˆ´Â f (1) = f (2), Rolleâ€™s Theorem applicable for function â€˜fâ€™ on [1, 2].

Letâ€™s find the derivative of f(x)

**(iv) f (x) = x (x â€“ 1) ^{2} on [0, 1]**

**Solution:**

Given function is f (x) = x(x â€“ 1)^{2}Â on [0, 1]

Since, given function f is a polynomial it is continuous and differentiable everywhere that is, on R.

Let us find the values at extremes

â‡’Â f (0) = 0 (0 â€“ 1)^{2}

â‡’Â f (0) = 0

â‡’Â f (1) = 1 (1 â€“ 1)^{2}

â‡’Â f (1) = 0^{2}

â‡’Â f (1) = 0

âˆ´Â f (0) = f (1), Rolleâ€™s theorem applicable for function â€˜fâ€™ on [0,1].

Letâ€™s find the derivative of f(x)

**(v) f (x) = (x ^{2 }â€“ 1) (x â€“ 2) on [-1, 2]**

**Solution:**

Given function is f (x) = (x^{2}Â â€“ 1) (x â€“ 2) on [â€“ 1, 2]

Since, given function f is a polynomial it is continuous and differentiable everywhere that is on R.

Let us find the values at extremes:

â‡’Â f ( â€“ 1) = (( â€“ 1)^{2}Â â€“ 1)( â€“ 1 â€“ 2)

â‡’Â f ( â€“ 1) = (1 â€“ 1)( â€“ 3)

â‡’Â f ( â€“ 1) = (0)( â€“ 3)

â‡’Â f ( â€“ 1) = 0

â‡’Â f (2) = (2^{2}Â â€“ 1)(2 â€“ 2)

â‡’Â f (2) = (4 â€“ 1)(0)

â‡’Â f (2) = 0

âˆ´Â f (â€“ 1) = f (2), Rolleâ€™s theorem applicable for function f on [ â€“ 1,2].

Letâ€™s find the derivative of f(x)

**(vi) f (x) = x (x â€“ 4) ^{2} on [0, 4]**

**Solution:**

Given function is f (x) = x (x â€“ 4)^{2}Â on [0, 4]

Since, given function f is a polynomial it is continuous and differentiable everywhere i.e., on R.

Let us find the values at extremes:

â‡’Â f (0) = 0(0 â€“ 4)^{2}

â‡’Â f (0) = 0

â‡’Â f (4) = 4(4 â€“ 4)^{2}

â‡’Â f (4) = 4(0)^{2}

â‡’Â f (4) = 0

âˆ´Â f (0) = f (4), Rolleâ€™s theorem applicable for function â€˜fâ€™ on [0,4].

Letâ€™s find the derivative of f(x):

**(vii) f (x) = x (x â€“ 2) ^{2} on [0, 2]**

**Solution:**

Given function is f (x) = x (x â€“ 2)^{2}Â on [0, 2]

Since, given function f is a polynomial it is continuous and differentiable everywhere that is on R.

Let us find the values at extremes:

â‡’Â f (0) = 0(0 â€“ 2)^{2}

â‡’Â f (0) = 0

â‡’Â f (2) = 2(2 â€“ 2)^{2}

â‡’Â f (2) = 2(0)^{2}

â‡’Â f (2) = 0

f (0) = f(2), Rolleâ€™s theorem applicable for function f on [0,2].

Letâ€™s find the derivative of f(x)

**(viii) f (x) = x ^{2} + 5x + 6 on [-3, -2]**

**Solution:**

Given function is f (x) = x^{2}Â + 5x + 6 on [â€“ 3, â€“ 2]

Since, given function f is a polynomial it is continuous and differentiable everywhere i.e., on R. Let us find the values at extremes:

â‡’Â f ( â€“ 3) = ( â€“ 3)^{2}Â + 5( â€“ 3) + 6

â‡’Â f ( â€“ 3) = 9 â€“ 15 + 6

â‡’Â f ( â€“ 3) = 0

â‡’Â f ( â€“ 2) = ( â€“ 2)^{2}Â + 5( â€“ 2) + 6

â‡’Â f ( â€“ 2) = 4 â€“ 10 + 6

â‡’Â f ( â€“ 2) = 0

âˆ´Â f (â€“ 3) = f( â€“ 2), Rolleâ€™s theorem applicable for function f on [ â€“ 3, â€“ 2].

Letâ€™s find the derivative of f(x):

**3. Verify the Rolleâ€™s Theorem for each of the following functions on the indicated intervals:**

**(i) f (x) = cos 2 (x â€“ Ï€/4) on [0, Ï€/2]**

**Solution:**

**(ii) f (x) = sin 2x on [0, Ï€/2]**

**Solution:**

**(iii) f (x) = cos 2x on [-Ï€/4, Ï€/4]**

**Solution:**

**(iv) f (x) = e ^{x} sin x on [0, Ï€]**

**Solution:**

Given function is f (x) = e^{x }sin x on [0, Ï€]

We know that exponential and sine functions are continuous and differentiable on R. Letâ€™s find the values of the function at an extreme,

**(v) f (x) = e ^{x} cos x on [-Ï€/2, Ï€/2]**

**Solution:**

**(vi) f (x) = cos 2x on [0, Ï€]**

**Solution:**

Given function is f (x) = cos 2x on [0, Ï€]

We know that cosine function is continuous and differentiable on R. Letâ€™s find the values of function at extreme,

â‡’Â f (0) = cos2(0)

â‡’Â f (0) = cos(0)

â‡’Â f (0) = 1

â‡’Â f (Ï€) = cos2(Ï€)

â‡’Â f (Ï€) = cos(2 Ï€)

â‡’Â f (Ï€) = 1

We have f (0) = f (Ï€), so there exist a c belongs to (0, Ï€) such that fâ€™(c) = 0.

Letâ€™s find the derivative of f(x)

Hence Rolleâ€™s Theorem is verified.

**Solution:**

**(viii) f (x) = sin 3x on [0, Ï€]**

**Solution:**

Given function is f (x) = sin3x on [0, Ï€]

We know that sine function is continuous and differentiable on R. Letâ€™s find the values of function at extreme,

â‡’Â f (0) = sin3(0)

â‡’Â f (0) = sin0

â‡’Â f (0) = 0

â‡’Â f (Ï€) = sin3(Ï€)

â‡’Â f (Ï€) = sin(3 Ï€)

â‡’Â f (Ï€) = 0

We have f (0) = f (Ï€), so there exist a c belongs to (0, Ï€) such that fâ€™(c) = 0.

Letâ€™s find the derivative of f(x)

**Solution:**

**(x) f (x) = log (x ^{2} + 2) â€“ log 3 on [-1, 1]**

**Solution:**

Given function is f (x) = log(x^{2}Â + 2) â€“ log3 on [â€“ 1, 1]

We know that logarithmic function is continuous and differentiable in its own domain. We check the values of the function at the extreme,

â‡’Â f (â€“ 1) = log((â€“ 1)^{2}Â + 2) â€“ log 3

â‡’Â f (â€“ 1) = log (1 + 2) â€“ log 3

â‡’Â f (â€“ 1) = log 3 â€“ log 3

â‡’Â f ( â€“ 1) = 0

â‡’Â f (1) = log (1^{2}Â + 2) â€“ log 3

â‡’Â f (1) = log (1 + 2) â€“ log 3

â‡’Â f (1) = log 3 â€“ log 3

â‡’Â f (1) = 0

We have got f (â€“ 1) = f (1). So, there exists a c such that c Ïµ (â€“ 1, 1) such that fâ€™(c) = 0.

Letâ€™s find the derivative of the function f,

**(xi) f (x) = sin x + cos x on [0, Ï€/2]**

**Solution:**

**(xii) f (x) = 2 sin x + sin 2x on [0, Ï€]**

**Solution:**

Given function is f (x) = 2sinx + sin2x on [0, Ï€]

We know that sine function continuous and differentiable over R.

Letâ€™s check the values of function f at the extremes

â‡’Â f (0) = 2sin(0) + sin2(0)

â‡’Â f (0) = 2(0) + 0

â‡’Â f (0) = 0

â‡’Â f (Ï€) = 2sin(Ï€) + sin2(Ï€)

â‡’Â f (Ï€) = 2(0) + 0

â‡’Â f (Ï€) = 0

We have f (0) = f (Ï€), so there exist a c belongs to (0, Ï€) such that fâ€™(c) = 0.

Letâ€™s find the derivative of function f.

â‡’Â fâ€™(x) = 2cosx + 2cos2x

â‡’Â fâ€™(x) = 2cosx + 2(2cos^{2}x â€“ 1)

â‡’Â fâ€™(x) = 4 cos^{2}x + 2 cos x â€“ 2

We have fâ€™(c) = 0,

â‡’Â 4cos^{2}c + 2 cos c â€“ 2 = 0

â‡’Â 2cos^{2}c + cos c â€“ 1 = 0

â‡’Â 2cos^{2}c + 2 cos c â€“ cos c â€“ 1 = 0

â‡’Â 2 cos c (cos c + 1) â€“ 1 (cos c + 1) = 0

â‡’Â (2cos c â€“ 1) (cos c + 1) = 0

**Solution:**

**Solution:**

**(xv) f (x) = 4 ^{sin x} on [0, Ï€]**

**Solution:**

**(xvi) f (x) = x ^{2} â€“ 5x + 4 on [0, Ï€/6]**

**Solution:**

Given function is f (x) = x^{2}Â â€“ 5x + 4 on [1, 4]

Since, given function f is a polynomial it is continuous and differentiable everywhere i.e., on R.

Let us find the values at extremes

â‡’Â f (1) = 1^{2}Â â€“ 5(1) + 4

â‡’Â f (1) = 1 â€“ 5 + 4

â‡’Â f (1) = 0

â‡’Â f (4) = 4^{2}Â â€“ 5(4) + 4

â‡’Â f (4) = 16 â€“ 20 + 4

â‡’Â f (4) = 0

We have f (1) = f (4). So, there exists a c Ïµ (1, 4) such that fâ€™(c) = 0.

Letâ€™s find the derivative of f(x):

**(xvii) f (x) = sin ^{4} x + cos^{4} x on [0, Ï€/2]**

**Solution:**

**(xviii) f (x) = sin x â€“ sin 2x on [0, Ï€]**

**Solution:**

Given function is f (x) = sin x â€“ sin2x on [0, Ï€]

We know that sine function is continuous and differentiable over R.

Now we have to check the values of the function â€˜fâ€™ at the extremes.

â‡’Â f (0) = sin (0)â€“sin 2(0)

â‡’Â f (0) = 0 â€“ sin (0)

â‡’Â f (0) = 0

â‡’Â f (Ï€) = sin(Ï€) â€“ sin2(Ï€)

â‡’Â f (Ï€) = 0 â€“ sin(2Ï€)

â‡’Â f (Ï€) = 0

We have f (0) = f (Ï€). So, there exists a c Ïµ (0, Ï€)Â such that fâ€™(c) = 0.

Now we have to find the derivative of the function â€˜fâ€™

**4. Using Rolleâ€™s Theorem, find points on the curve y = 16 â€“ x ^{2}, x âˆˆ [-1, 1], where tangent is parallel to x â€“ axis.**

**Solution:**

Given function is y = 16 â€“ x^{2}, x Ïµ [â€“ 1, 1]

We know that polynomial function is continuous and differentiable over R.

Let us check the values of â€˜yâ€™ at extremes

â‡’Â y (â€“ 1) = 16 â€“ (â€“ 1)^{2}

â‡’Â y (â€“ 1) = 16 â€“ 1

â‡’Â y (â€“ 1) = 15

â‡’Â y (1) = 16 â€“ (1)^{2}

â‡’Â y (1) = 16 â€“ 1

â‡’Â y (1) = 15

We have y (â€“ 1) = y (1). So, there exists a c Ïµ (â€“ 1, 1) such that fâ€™(c) = 0.

We know that for a curve g, the value of the slope of the tangent at a point r is given by gâ€™(r).

Now we have to find the derivative of curve y

â‡’Â

â‡’Â yâ€™ = â€“ 2x

We have yâ€™(c) = 0

â‡’Â â€“ 2c = 0

â‡’Â c = 0 Ïµ (â€“ 1, 1)

Value of y at x = 1 is

â‡’Â y = 16 â€“ 0^{2}

â‡’Â y = 16

âˆ´Â The point at which the curve y has a tangent parallel to x â€“ axis (since the slope of x â€“ axis is 0) is (0, 16).

Exercise 15.2 Page No: 15.17

**1. Verify Lagrangeâ€™s mean value theorem for the following functions on the indicated intervals. In each case find a point â€˜câ€™ in the indicated interval as stated by the Lagrangeâ€™s mean value theorem:**

**(i) f (x) = x ^{2} â€“ 1 on [2, 3]**

**Solution:**

**(ii) f (x) = x ^{3} â€“ 2x^{2} â€“ x + 3 on [0, 1]**

**Solution:**

Given f (x) = x^{3}Â â€“ 2x^{2}Â â€“ x + 3 on [0, 1]

Every polynomial function isÂ continuousÂ everywhere on (âˆ’âˆž, âˆž) andÂ differentiableÂ for all arguments. Here, f(x) is a polynomial function. So it is continuous in [0, 1] and differentiable in (0, 1). So both the necessary conditions ofÂ Lagrangeâ€™s mean value theorem is satisfied.

Therefore, there exist a point c âˆˆ (0, 1) such that:

f (x) = x^{3}Â â€“ 2x^{2}Â â€“ x + 3

Differentiating with respect to x

fâ€™(x) = 3x^{2}Â â€“ 2(2x) â€“ 1

= 3x^{2}Â â€“ 4x â€“ 1

For fâ€™(c), put the value of x=c in fâ€™(x)

fâ€™(c)= 3c^{2}Â â€“ 4c â€“ 1

For f (1), put the value of x = 1 in f(x)

f (1)= (1)^{3}Â â€“ 2(1)^{2}Â â€“ (1) + 3

= 1 â€“ 2 â€“ 1 + 3

= 1

For f (0), put the value of x=0 in f(x)

f (0)= (0)^{3}Â â€“ 2(0)^{2}Â â€“ (0) + 3

= 0 â€“ 0 â€“ 0 + 3

= 3

âˆ´Â fâ€™(c) = f(1) â€“ f(0)

â‡’Â 3c^{2}Â â€“ 4c â€“ 1 = 1 â€“ 3

â‡’Â 3c^{2}Â â€“ 4c = 1 + 1 â€“ 3

â‡’Â 3c^{2}Â â€“ 4c = â€“ 1

â‡’Â 3c^{2}Â â€“ 4c + 1 = 0

â‡’Â 3c^{2}Â â€“ 3c â€“ c + 1 = 0

â‡’Â 3c(c â€“ 1) â€“ 1(c â€“ 1) = 0

â‡’Â (3c â€“ 1) (c â€“ 1) = 0

**(iii) f (x) = x (x â€“ 1) on [1, 2]**

**Solution:**

Given f (x) = x (x â€“ 1) on [1, 2]

= x^{2}Â â€“ x

Every polynomial function isÂ continuousÂ everywhere on (âˆ’âˆž, âˆž) andÂ differentiableÂ for all arguments. Here, f(x) is a polynomial function. So it is continuous in [1, 2] and differentiable in (1, 2). So both the necessary conditions ofÂ Lagrangeâ€™s mean value theorem is satisfied.

Therefore, there exist a point c âˆˆ (1, 2) such that:

f (x) = x^{2}Â â€“ x

Differentiating with respect to x

fâ€™(x) = 2x â€“ 1

For fâ€™(c), put the value of x=c in fâ€™(x):

fâ€™(c)= 2c â€“ 1

For f (2), put the value of x = 2 in f(x)

f (2) = (2)^{2}Â â€“ 2

= 4 â€“ 2

= 2

For f (1), put the value of x = 1 in f(x):

f (1)= (1)^{2}Â â€“ 1

= 1 â€“ 1

= 0

âˆ´Â fâ€™(c) = f(2) â€“ f(1)

â‡’Â 2c â€“ 1 = 2 â€“ 0

â‡’Â 2c = 2 + 1

â‡’Â 2c = 3

**(iv) f (x) = x ^{2} â€“ 3x + 2 on [-1, 2]**

**Solution:**

Given f (x) = x^{2}Â â€“ 3x + 2 on [â€“ 1, 2]

Every polynomial function isÂ continuousÂ everywhere on (âˆ’âˆž, âˆž) andÂ differentiableÂ for all arguments. Here, f(x) is a polynomial function. So it is continuous in [â€“ 1, 2] and differentiable in (â€“ 1, 2). So both the necessary conditions ofÂ Lagrangeâ€™s mean value theorem is satisfied.

Therefore, there exist a point c âˆˆ (-1, 2) such that:

f (x) = x^{2}Â â€“ 3x + 2

Differentiating with respect to x

fâ€™(x) = 2x â€“ 3

For fâ€™(c), put the value of x = c in fâ€™(x):

fâ€™(c)= 2c â€“ 3

For f (2), put the value of x = 2 in f(x)

f (2) = (2)^{2}Â â€“ 3 (2) + 2

= 4 â€“ 6 + 2

= 0

For f (â€“ 1), put the value of x = â€“ 1 in f(x):

f (â€“ 1) = (â€“ 1)^{2}Â â€“ 3 (â€“ 1) + 2

= 1 + 3 + 2

= 6

**(v) f (x) = 2x ^{2} â€“ 3x + 1 on [1, 3]**

**Solution:**

Given f (x) = 2x^{2}Â â€“ 3x + 1 on [1, 3]

Every polynomial function isÂ continuousÂ everywhere on (âˆ’âˆž, âˆž) andÂ differentiableÂ for all arguments. Here, f(x) is a polynomial function. So it is continuous in [1, 3] and differentiable in (1, 3). So both the necessary conditions ofÂ Lagrangeâ€™s mean value theorem is satisfied.

Therefore, there exist a point c âˆˆ (1, 3) such that:

f (x) = 2x^{2}Â â€“ 3x + 1

Differentiating with respect to x

fâ€™(x) = 2(2x) â€“ 3

= 4x â€“ 3

For fâ€™(c), put the value of x = c in fâ€™(x):

fâ€™(c)= 4c â€“ 3

For f (3), put the value of x = 3 in f(x):

f (3) = 2 (3)^{2}Â â€“ 3 (3) + 1

= 2 (9) â€“ 9 + 1

= 18 â€“ 8 = 10

For f (1), put the value of x = 1 in f(x):

f (1) = 2 (1)^{2}Â â€“ 3 (1) + 1

= 2 (1) â€“ 3 + 1

= 2 â€“ 2 = 0

**(vi) f (x) = x ^{2} â€“ 2x + 4 on [1, 5]**

**Solution:**

Given f (x) = x^{2}Â â€“ 2x + 4 on [1, 5]

Every polynomial function isÂ continuousÂ everywhere on (âˆ’âˆž, âˆž) andÂ differentiableÂ for all arguments. Here, f(x) is a polynomial function. So it is continuous in [1, 5] and differentiable in (1, 5). So both the necessary conditions ofÂ Lagrangeâ€™s mean value theorem is satisfied.

Therefore, there exist a point c âˆˆ (1, 5) such that:

f (x) = x^{2}Â â€“ 2x + 4

Differentiating with respect to x:

fâ€™(x) = 2x â€“ 2

For fâ€™(c), put the value of x=c in fâ€™(x):

fâ€™(c)= 2c â€“ 2

For f (5), put the value of x=5 in f(x):

f (5)= (5)^{2}Â â€“ 2(5) + 4

= 25 â€“ 10 + 4

= 19

For f (1), put the value of x = 1 in f(x)

f (1) = (1)^{2}Â â€“ 2 (1) + 4

= 1 â€“ 2 + 4

= 3

**(vii) f (x) = 2x â€“ x ^{2} on [0, 1]**

**Solution:**

Given f (x) = 2x â€“ x^{2}Â on [0, 1]

Every polynomial function isÂ continuousÂ everywhere on (âˆ’âˆž, âˆž) andÂ differentiableÂ for all arguments. Here, f(x) is a polynomial function. So it is continuous in [0, 1] and differentiable in (0, 1). So both the necessary conditions ofÂ Lagrangeâ€™s mean value theorem is satisfied.

Therefore, there exist a point c âˆˆ (0, 1) such that:

â‡’Â fâ€™(c) = f(1) â€“ f(0)

f (x) = 2x â€“ x^{2}

Differentiating with respect to x:

fâ€™(x) = 2 â€“ 2x

For fâ€™(c), put the value of x = c in fâ€™(x):

fâ€™(c)= 2 â€“ 2c

For f (1), put the value of x = 1 in f(x):

f (1)= 2(1) â€“ (1)^{2}

= 2 â€“ 1

= 1

For f (0), put the value of x = 0 in f(x):

f (0) = 2(0) â€“ (0)^{2}

= 0 â€“ 0

= 0

fâ€™(c) = f(1) â€“ f(0)

â‡’Â 2 â€“ 2c = 1 â€“ 0

â‡’Â â€“ 2c = 1 â€“ 2

â‡’Â â€“ 2c = â€“ 1

**(viii) f (x) = (x â€“ 1) (x â€“ 2) (x â€“ 3)**

**Solution:**

Given f (x) = (x â€“ 1) (x â€“ 2) (x â€“ 3) on [0, 4]

= (x^{2}Â â€“ x â€“ 2x + 3) (x â€“ 3)

= (x^{2}Â â€“ 3x + 3) (x â€“ 3)

= x^{3}Â â€“ 3x^{2}Â + 3x â€“ 3x^{2}Â + 9x â€“ 9

= x^{3}Â â€“ 6x^{2}Â + 12x â€“ 9 on [0, 4]

Every polynomial function isÂ continuousÂ everywhere on (âˆ’âˆž, âˆž) andÂ differentiableÂ for all arguments. Here, f(x) is a polynomial function. So it is continuous in [0, 4] and differentiable in (0, 4). So both the necessary conditions ofÂ Lagrangeâ€™s mean value theorem is satisfied.

Therefore, there exist a point c âˆˆ (0, 4) such that:

f (x) = x^{3}Â â€“ 6x^{2}Â + 12x â€“ 9

Differentiating with respect to x:

fâ€™(x) = 3x^{2}Â â€“ 6(2x) + 12

= 3x^{2}Â â€“ 12x + 12

For fâ€™(c), put the value of x = c in fâ€™(x):

fâ€™(c)= 3c^{2}Â â€“ 12c + 12

For f (4), put the value of x = 4 in f(x):

f (4)= (4)^{3}Â â€“ 6(4)^{2}Â + 12 (4) â€“ 9

= 64 â€“ 96 + 48 â€“ 9

= 7

For f (0), put the value of x = 0 in f(x):

f (0)= (0)^{3}Â â€“ 6 (0)^{2}Â + 12 (0) â€“ 9

= 0 â€“ 0 + 0 â€“ 9

= â€“ 9

**Solution:**

**(x) f (x) = tan ^{-1}x on [0, 1]**

**Solution:**

Given f (x) = tanÂ ^{â€“ 1}Â x on [0, 1]

TanÂ ^{â€“ 1}Â x has unique value for all x between 0 and 1.

âˆ´Â f (x) is continuous in [0, 1]

f (x) = tanÂ ^{â€“ 1}Â x

Differentiating with respect to x:

x^{2}Â always has value greater than 0.

â‡’Â 1 + x^{2}Â > 0

âˆ´Â f (x) is differentiable in (0, 1)

So both the necessary conditions ofÂ Lagrangeâ€™s mean value theorem is satisfied. Therefore, there exist a point c âˆˆ (0, 1) such that:

**Solution:**

**(xii) f (x) = x (x + 4) ^{2} on [0, 4]**

**Solution:**

Given f (x) = x (x + 4)^{2}Â on [0, 4]

= x [(x)^{2 }+ 2 (4) (x) + (4)^{2}]

= x (x^{2}Â + 8x + 16)

= x^{3}Â + 8x^{2}Â + 16x on [0, 4]

Every polynomial function isÂ continuousÂ everywhere on (âˆ’âˆž, âˆž) andÂ differentiableÂ for all arguments. Here, f(x) is a polynomial function. So it is continuous in [0, 4] and differentiable in (0, 4). So both the necessary conditions ofÂ Lagrangeâ€™s mean value theorem is satisfied. Therefore, there exist a point c âˆˆ (0, 4) such that:

f (x) = x^{3}Â + 8x^{2}Â + 16x

Differentiating with respect to x:

fâ€™(x) = 3x^{2}Â + 8(2x) + 16

= 3x^{2}Â + 16x + 16

For fâ€™(c), put the value of x = c in fâ€™(x):

fâ€™(c)= 3c^{2}Â + 16c + 16

For f (4), put the value of x = 4 in f(x):

f (4)= (4)^{3}Â + 8(4)^{2}Â + 16(4)

= 64 + 128 + 64

= 256

For f (0), put the value of x = 0 in f(x):

f (0)= (0)^{3}Â + 8(0)^{2}Â + 16(0)

= 0 + 0 + 0

= 0

**Solution:**

**(xiv) f (x) = x ^{2} + x â€“ 1 on [0, 4]**

**Solution:**

Given f (x) = x^{2}Â + x â€“ 1 on [0, 4]

Every polynomial function isÂ continuousÂ everywhere on (âˆ’âˆž, âˆž) andÂ differentiableÂ for all arguments. Here, f(x) is a polynomial function. So it is continuous in [0, 4] and differentiable in (0, 4). So both the necessary conditions ofÂ Lagrangeâ€™s mean value theorem is satisfied. Therefore, there exist a point c âˆˆ (0, 4) such that:

f (x) = x^{2}Â + x â€“ 1

Differentiating with respect to x:

fâ€™(x) = 2x + 1

For fâ€™(c), put the value of x = c in fâ€™(x):

fâ€™(c) = 2c + 1

For f (4), put the value of x = 4 in f(x):

f (4)= (4)^{2}Â + 4 â€“ 1

= 16 + 4 â€“ 1

= 19

For f (0), put the value of x = 0 in f(x):

f (0) = (0)^{2}Â + 0 â€“ 1

= 0 + 0 â€“ 1

= â€“ 1

**(xv) f (x) = sin x â€“ sin 2x â€“ x on [0, Ï€]**

**Solution:**

Given f (x) = sin x â€“ sin 2x â€“ x on [0, Ï€]

Sin x and cos x functions areÂ continuousÂ everywhere on (âˆ’âˆž, âˆž) andÂ differentiableÂ for all arguments. So both the necessary conditions ofÂ Lagrangeâ€™s mean value theorem is satisfied. Therefore, there exist a point c âˆˆ (0, Ï€) such that:

**(xvi) f (x) = x ^{3} â€“ 5x^{2} â€“ 3x on [1, 3]**

**Solution:**

Given f (x) = x^{3}Â â€“ 5x^{2}Â â€“ 3x on [1, 3]

Every polynomial function isÂ continuousÂ everywhere on (âˆ’âˆž, âˆž) andÂ differentiableÂ for all arguments. Here, f(x) is a polynomial function. So it is continuous in [1, 3] and differentiable in (1, 3). So both the necessary conditions ofÂ Lagrangeâ€™s mean value theorem is satisfied.

Therefore, there exist a point c âˆˆ (1, 3) such that:

f (x) = x^{3}Â â€“ 5x^{2}Â â€“ 3x

Differentiating with respect to x:

fâ€™(x) = 3x^{2}Â â€“ 5(2x) â€“ 3

= 3x^{2}Â â€“ 10x â€“ 3

For fâ€™(c), put the value of x=c in fâ€™(x):

fâ€™(c)= 3c^{2}Â â€“ 10c â€“ 3

For f (3), put the value of x = 3 in f(x):

f (3)= (3)^{3}Â â€“ 5(3)^{2}Â â€“ 3(3)

= 27 â€“ 45 â€“ 9

= â€“ 27

For f (1), put the value of x = 1 in f(x):

f (1)= (1)^{3}Â â€“ 5 (1)^{2}Â â€“ 3 (1)

= 1 â€“ 5 â€“ 3

= â€“ 7

â‡’Â 3c^{2}Â â€“ 10c â€“ 3 = â€“ 10

â‡’Â 3c^{2}Â â€“ 10c â€“ 3 + 10 = 0

â‡’Â 3c^{2}Â â€“ 10c + 7 = 0

â‡’Â 3c^{2}Â â€“ 7c â€“ 3c + 7 = 0

â‡’Â c (3c â€“ 7) â€“ 1(3c â€“ 7) = 0

â‡’Â (3c â€“ 7) (c â€“ 1) = 0

**2. Discuss the applicability of Lagrangeâ€™s mean value theorem for the function f(x) = |x| on [â€“ 1, 1].**

**Solution:**

**3. Show that the Lagrange’s mean value theorem is not applicable to the function f(x) = 1/x on [â€“1, 1].**

**Solution:**

Given

Here, x â‰ 0

â‡’Â f (x) exists for all values of x except 0

â‡’Â f (x) is discontinuous at x=0

âˆ´Â f (x) is not continuous in [â€“ 1, 1]

Hence the Lagrangeâ€™s mean value theorem is not applicable to the function f (x) = 1/x on [-1, 1]

**4. Verify the hypothesis and conclusion of Lagrangeâ€™s mean value theorem for the function**

**Solution:**

Given

**5. Find a point on the parabola y = (x â€“ 4) ^{2}, where the tangent is parallel to the chord joining (4, 0) and (5, 1).**

**Solution:**

Given f(x) = (x â€“ 4)^{2}Â on [4, 5]

This interval [a, b] is obtained by x â€“ coordinates of the points of the chord.

Every polynomial function isÂ continuousÂ everywhere on (âˆ’âˆž, âˆž) andÂ differentiableÂ for all arguments. Here, f(x) is a polynomial function. So it is continuous in [4, 5] and differentiable in (4, 5). So both the necessary conditions ofÂ Lagrangeâ€™s mean value theorem is satisfied.

Therefore, there exist a point c âˆˆ (4, 5) such that:

â‡’Â fâ€™(x) = 2 (x â€“ 4) (1)

â‡’Â fâ€™(x) = 2 (x â€“ 4)

For fâ€™(c), put the value of x=c in fâ€™(x):

fâ€™(c) = 2 (c â€“ 4)

For f (5), put the value of x=5 in f(x):

f (5) = (5 â€“ 4)^{2}

= (1)^{2}

= 1

For f (4), put the value of x=4 in f(x):

f (4) = (4 â€“ 4)^{2}

= (0)^{2}

= 0

fâ€™(c) = f(5) â€“ f(4)

â‡’Â 2(c â€“ 4) =Â 1 â€“ 0

â‡’Â 2c â€“ 8 =Â 1

â‡’Â 2c =Â 1 + 8

We know that, the value of c obtained in Lagrangeâ€™s Mean value Theorem is nothing but the value of x â€“ coordinate of the point of the contact of the tangent to the curve which is parallel to the chord joining the points (4, 0) and (5, 1).

Now, put this value of x in f(x) to obtain y:

y = (x â€“ 4)^{2}