NCERT solutions for Class 12 Mathematics Chapter 6 Applications of Derivatives – complete answers & explanations

Class 12 Mathematics Chapter 6 Applications of Derivatives is an important chapter that helps students understand how derivatives are used in real-life situations. This chapter focuses on concepts like rate of change, increasing and decreasing functions, maxima and minima, and optimization problems. These topics are essential for solving practical problems and are frequently asked in board exams. This blog provides clear and reliable NCERT solutions that help students understand the correct approach to answering questions. Students can improve their problem-solving skills and build strong concepts by practicing regularly. Download the worksheet and practice alongside solutions for better clarity. With proper guidance and practice, students can gain confidence in this chapter. Book a free trial now to get expert guidance.

What this NCERT chapter covers?

1. Understanding how derivatives are applied to solve real-life problems involving change and motion.  
2. Learning about rate of change of quantities such as area, volume, and distance.  
3. Studying increasing and decreasing functions using derivatives.  
4. Identifying intervals where functions behave differently.  
5. Finding maxima and minima values of functions.  
6. Solving optimization problems related to profit, cost, and measurements.  
7. Understanding critical points and their importance in graph behavior.  
8. Applying second derivative tests for maxima and minima.  
9. Interpreting results in practical contexts for better understanding.  
10. Strengthening problem-solving skills required for exams.  

How to use these NCERT solutions?

1. Start by attempting each question from the worksheet on your own before checking the answers.  
2. Carefully compare your answers with the given solutions to identify mistakes.  
3. Focus on understanding where you went wrong instead of just copying answers.  
4. Practice similar types of questions to improve accuracy and speed.  
5. Follow the exact order of questions to stay aligned with the worksheet.  
6. Use the solutions to revise important concepts before exams.  
7. Parents and teachers can guide students by checking their approach step-by-step.  
8. Regular practice using these solutions helps build confidence and clarity.  

Important tips & tricks for students

1. Always understand the concept before solving derivative-based problems.  
2. Be careful while differentiating functions to avoid calculation errors.  
3. Pay attention to signs while identifying increasing and decreasing intervals.  
4. Practice maxima and minima problems regularly to master them.  
5. Write steps clearly to score full marks in exams.  
6. Revise formulas related to derivatives frequently.  
7. Double-check answers, especially in word problems.  
8. Stay consistent with practice to improve speed and accuracy.  

NCERT solutions – complete answer key

Exercise No. 6.1

1. (a) 6π  
  (b) 8π  
2. 48 cm²/s  
3. 60π cm²/s  
4. 3000 cm³/s  
5. 80π cm²/s  
6. 4/π cm/s  
7. 1.4π cm/s  
8. (a) –2 cm/min  
  (b) –2 cm²/min  
9. –8/3 cm/s  
10. 200π cm³ per unit increase in radius  
11. 2π cm³/s  
12. (2, 2) and (–2, –2)  
13. π/2 (2x + 1)²  
14. 1/24 cm/s  
15. 18.986 ≈ 18.99  
16. 208  
17. 12π  
18. 126  

Exercise No. 6.2

1. Increasing on R  
2. Increasing on R  
3. (a) Increasing in (0, π/2)  
  (b) Decreasing in (π/2, π)  
  (c) Neither in (0, π)  
4. (a) Increasing in (3/4, ∞)  
  (b) Decreasing in (–∞, 3/4)  
5. (a) Increasing in (–∞, –2) ∪ (3, ∞)  
  (b) Decreasing in (–2, 3)  
6. (a) Increasing: (–1, ∞), Decreasing: (–∞, –1)  
  (b) Increasing: (–∞, –3/2), Decreasing: (–3/2, ∞)  
  (c) Increasing: none, Decreasing: (–∞, –2) ∪ (–2, 0)  
  (d) Increasing: (–∞, –9/2), Decreasing: (–9/2, ∞)  
  (e) Increasing: (–∞, ∞)  
7. Increasing for x < 0 and x > 2  
8. Increasing on (–1, ∞)  
9. Increasing on (0, ∞)  
10. Increasing in (0, π/2)  
11. cos x  
12. Neither increasing nor decreasing  
13. a ≥ –3  
14. None  
15. d²y/dx² = 49y  
16. d²y/dx² = (dy/dx)²  
17. (x² + 1)² y₂ + 2x(x² + 1)y₁ = 2  
18. Increasing in R  
19. (D) (0, 2)  

Exercise No. 6.3

1. Maximum value = 4; Minimum value = 0  

2.  (i) Minimum = 3 at x = 1/2; No maximum  
(ii) Minimum = –2 at x = –2/3; No maximum  
(iii) Maximum = 10 at x = 1; No minimum  
(iv) No maximum, No minimum  

3.  (i) Minimum = –1 at x = –2; No maximum  
(ii) Maximum = 3 at x = –1; No minimum  
(iii) Maximum = 6, Minimum = 4  
(iv) Minimum = 0, Maximum = 4  
(v) No maximum, No minimum  

4.  (i) Minimum at x = 0, value = 0; No maximum  
(ii) Local max at x = –1, value = 2; Local min at x = 1, value = –2  
(iii) Local max at x = π/4, value = √2; Local min at x = 5π/4, value = –√2  
(iv) Local max at x = 3π/4; Local min at x = 7π/4  
(v) Local max at x = 1, value = 19; Local min at x = 3, value = 15  
(vi) Minimum at x = 1, value = 2  
(vii) Minimum at x = 0, value = 1/2  
(viii) Local max at x = 1/2, value = 1/4  

5.  (i) No maxima, No minima  
(ii) No maxima, No minima  
(iii) No maxima, No minima  

6. Maximum profit = 113  

7. Maximum value = 1; Minimum value = –1  

8. Maximum value = 3/2; Minimum value = –3/2  

9. Maximum value = 1; Minimum value = 0  

10. Maximum value = 1; Minimum value = 0  

11. Maximum value = 2; Minimum value = 0  

12. Maximum value = 4; Minimum value = –4  

13. Maximum value = 1; Minimum value = –1  

14. Maximum value = 27; Minimum value = 0  

15.  (i) Max = 8 at x = 2; Min = –8 at x = –2  
(ii) Max = √2 at x = π/4; Min = –√2 at x = 5π/4  
(iii) Max = 9/4; Min = 0  
(iv) Max = 4; Min = 0  

16. Maximum value = 1; Minimum value = 0  
17. Maximum value = 1; Minimum value = 0  
18. Maximum value = 2; Minimum value = 0  
19. Maximum value = 4; Minimum value = –4  
20. Maximum value = 1; Minimum value = –1  
21. Maximum value = 27; Minimum value = 0  
22. Maximum value = 1; Minimum value = 0  
23. Maximum value = 2; Minimum value = –216  
24. Maximum value = 1; Minimum value = –1  
25. Maximum value = 3; Minimum value = –3  
26. Maximum value = 1; Minimum value = 0  
27. Maximum value = 4; Minimum value = 0  
28. Maximum value = 9; Minimum value = 0  
29. Maximum value = 16; Minimum value = 0  

Miscellaneous Exercise on Chapter 6

1. Maximum at x = e  
2. Rate of decrease of area = (3√3 / 4) b² cm²/s  
3.  (i) Increasing in (−π/2, π/2)  
(ii) Decreasing in (π/2, 3π/2)  
4.  (i) Increasing in (−∞, −1) ∪ (1, ∞)  
(ii) Decreasing in (−1, 1)  
5. Maximum area = ab  
6. Least cost = Rs 520  
7. Minimum when side of square = 2r  
8. Radius = 5/π m, Height = 5/π m  
9. Minimum length = (a^(2/3) + b^(2/3))^(3/2)  
10.  Local maxima at x = –1, value = 0  
Local minima at x = 2, value = 0  
Point of inflexion at x = 1  
11. Maximum value = 5/4  
Minimum value = 3/4  
12. Altitude = √2 × radius  
13. Numbers = 12 and 12  
14. x = 15, y = 45  
15. x = 10, y = 25  
16. Numbers = 8 and 8  

Why NCERT solutions help students?

NCERT solutions help students prepare effectively for exams by providing the correct approach to solving questions. They improve concept clarity, strengthen problem-solving skills, and ensure students understand how to write accurate answers. With consistent practice, students can build confidence and perform better in their Class 12 Mathematics exams.

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