RD Sharma Solutions for Class 12 Maths Chapter 15 Mean Value Theorems – PDF Download

RD Sharma Solutions for Class 12 Maths Chapter 15 – Mean Value Theorems are provided here to help students solve exercise problems with ease and build a deeper understanding of the chapter. This chapter covers important theorems such as Rolle’s Theorem and the Mean Value Theorem, which are essential for both board exams and competitive tests like JEE.

These RD Sharma solutions serve as a reliable and comprehensive resource for students aiming to strengthen their mathematical foundation. Chapter 15 of RD Sharma Class 12 is known for its conceptual clarity and a wide range of problems for practice. Students can use these step-by-step solutions to revise efficiently and enhance their problem-solving speed and accuracy.

To support smooth learning and revision, we’ve made the RD Sharma Class 12 Chapter 15 Solutions available in a downloadable PDF format. These answers are presented in simple language and structured according to the latest CBSE guidelines, making them ideal for last-minute prep and self-study. Regular practice of these solutions not only ensures a solid grasp of theorems but also helps in understanding their practical applications—crucial for scoring full marks in board exams.

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

RD Sharma Class 12 Maths Solutions Chapter 15 Mean Value Theorems cover all the questions from the textbook, crafted by expert Mathematics teachers at Infinity Learn. Download our free PDF of Chapter 15 Mean Value Theorems RD Sharma Solutions for Class 12 to boost your performance in board exams and Entrance exams.

1. Verify Rolle’s Theorem for

f(x) = x² – 8x + 12 on [2, 6]

Solution:

Continuity & Differentiability: f(x) is a polynomial ⇒ continuous and differentiable everywhere.

Check f(2) = f(6):
f(2) = 4 – 16 + 12 = 0
f(6) = 36 – 48 + 12 = 0
⇒ f(2) = f(6)

Apply Rolle’s Theorem:
f'(x) = 2x – 8
Set f'(c) = 0 ⇒ 2c – 8 = 0 ⇒ c = 4
⇒ c ∈ (2, 6)

2. Verify Rolle’s Theorem for

f(x) = x² – 4x + 3 on [1, 3]

Solution:

Continuity & Differentiability: f(x) is a polynomial ⇒ continuous and differentiable.

Check f(1) = f(3):
f(1) = 1 – 4 + 3 = 0
f(3) = 9 – 12 + 3 = 0
⇒ f(1) = f(3)

Apply Rolle’s Theorem:
f'(x) = 2x – 4
Set f'(c) = 0 ⇒ 2c – 4 = 0 ⇒ c = 2
⇒ c ∈ (1, 3)

3. Verify Rolle’s Theorem for

f(x) = (x – 1)(x – 2)² on [1, 2]

Solution:

Continuity & Differentiability: f(x) is a polynomial ⇒ continuous and differentiable.

Check f(1) = f(2):
f(1) = 0, f(2) = 0 ⇒ f(1) = f(2)

Apply Rolle’s Theorem:
f'(x) = (x – 2)² + 2(x – 1)(x – 2)
Set f'(c) = 0
Solve for c in (1, 2)

4. Verify Rolle’s Theorem for

f(x) = x(x – 1)² on [0, 1]

Solution:

Continuity & Differentiability: f(x) is a polynomial ⇒ continuous and differentiable.

Check f(0) = f(1):
f(0) = 0, f(1) = 0 ⇒ f(0) = f(1)

Apply Rolle’s Theorem:
f'(x) = (x – 1)² + 2x(x – 1)
Set f'(c) = 0
Solve for c in (0, 1)

5. Verify Rolle’s Theorem for

f(x) = (x² – 1)(x – 2) on [–1, 2]

Solution:

Continuity & Differentiability: f(x) is a polynomial ⇒ continuous and differentiable.

Check f(–1) = f(2):
f(–1) = 0, f(2) = 0 ⇒ f(–1) = f(2)

Apply Rolle’s Theorem:
f'(x) = derivative of (x² – 1)(x – 2)
Set f'(c) = 0
Solve for c in (–1, 2)

6. Verify Rolle’s Theorem for

f(x) = x(x – 4)² on [0, 4]

Solution:

Continuity & Differentiability: f(x) is a polynomial ⇒ continuous and differentiable.

Check f(0) = f(4):
f(0) = 0, f(4) = 0 ⇒ f(0) = f(4)

Apply Rolle’s Theorem:
f'(x) = (x – 4)² + 2x(x – 4)
Set f'(c) = 0
Solve for c in (0, 4)

7. Verify Rolle’s Theorem for

f(x) = x(x – 2)² on [0, 2]

Solution:

Continuity & Differentiability: f(x) is a polynomial ⇒ continuous and differentiable.

Check f(0) = f(2):
f(0) = 0, f(2) = 0 ⇒ f(0) = f(2)

Apply Rolle’s Theorem:
f'(x) = (x – 2)² + 2x(x – 2)
Set f'(c) = 0
Solve for c in (0, 2)

8. Verify Rolle’s Theorem for

f(x) = x² + 5x + 6 on [–3, –2]

Solution:

Continuity & Differentiability: f(x) is a polynomial ⇒ continuous and differentiable.

Check f(–3) = f(–2):
f(–3) = 0, f(–2) = 0 ⇒ f(–3) = f(–2)

Apply Rolle’s Theorem:
f'(x) = 2x + 5
Set f'(c) = 0 ⇒ 2c + 5 = 0 ⇒ c = –2.5
⇒ c ∈ (–3, –2)

9. Verify Rolle’s Theorem for

f(x) = cos²(x – π/4) on [0, π/2]

Solution:

Continuity & Differentiability: f(x) is continuous and differentiable.

Check f(0) = f(π/2):
f(0) = cos²(–π/4) = 0.5
f(π/2) = cos²(π/4) = 0.5 ⇒ f(0) = f(π/2)

Apply Rolle’s Theorem:
f'(x) = –2cos(x – π/4)sin(x – π/4) = –sin(2x – π/2)
Set f'(c) = 0 ⇒ sin(2c – π/2) = 0 ⇒ 2c – π/2 = nπ ⇒ c = (π/2 + nπ)/2
Find c ∈ (0, π/2)

10. Verify Rolle’s Theorem for

f(x) = sin(2x) on [0, π/2]

Solution:

Continuity & Differentiability: f(x) is continuous and differentiable.

Check f(0) = f(π/2):
f(0) = 0, f(π/2) = sin(π) = 0 ⇒ f(0) = f(π/2)

Apply Rolle’s Theorem:
f'(x) = 2cos(2x)
Set f'(c) = 0 ⇒ cos(2c) = 0 ⇒ 2c = π/2 ⇒ c = π/4
⇒ c ∈ (0, π/2)