Exercise 2.7 Class 12 Math Solutions - Unit 2 Further Differentiation, Punjab Board

Step-by-step solutions to Exercise 2.7, Unit 2: Further Differentiation, for Class 12 Punjab Board students. Covers applied real-world optimization problems, formulating objective functions from geometric and economic constraints, finding critical dimensions, and verifying maxima and minima using the second derivative test, worked out question by question.

This page provides complete solved solutions for Exercise 2.7 of Unit 2: Further Differentiation, Class 12 Mathematics, for Punjab Board (PCTB / PECTAA). Every applied real-world optimization problem is solved step by step, guiding students through variable definitions, constraint equations, formulating single-variable objective functions, computing critical values, and confirming maxima or minima using the Second Derivative Test.
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Exercise Overview

Exercise 2.7 is the grand practical application concluding Unit 2 (Further Differentiation) in Class 12 Mathematics under the Punjab Board PCTB / PECTAA curriculum. This exercise bridges pure differential calculus with real-world optimization challenges in engineering, geometry, and economics. Students apply a rigorous 7-step analytical framework: Understanding the Problem, Defining Variables, Formulating the Objective Function, Expressing Constraints, Finding Critical Points, Verifying Extrema through the Second Derivative Test, and Interpreting Physical Feasibility.

The exercise features classic optimization scenarios, including maximizing the volume of an open-top box constructed from a sheet of cardboard, minimizing the total surface area (and manufacturing material cost) of a closed cylindrical can with fixed volume, maximizing fenced rectangular areas along river boundaries, and determining optimal economic pricing for maximum profit and revenue. Every problem teaches students how to eliminate secondary variables using constraint relationships to yield a single-variable function (f(x)) suitable for differentiation.

For Punjab Board HSSC Part 2 examinations across all nine educational divisions (Lahore, Rawalpindi, Gujranwala, Faisalabad, Multan, Bahawalpur, Sargodha, Sahiwal, and D.G. Khan), Exercise 2.7 is universally recognized as one of the most prominent sources of Section C calculus Long Questions. These step-by-step solutions provide clear diagrams, explicit constraint substitutions, and complete second derivative concavity checks for every problem.

FAQs

Are these Exercise 2.7 solutions according to the Punjab Board syllabus?
Yes. These solutions follow the Punjab Curriculum and Textbook Board (PCTB / PECTAA) syllabus for Class 12 Mathematics, matching the applied optimization problems in Unit 2, Exercise 2.7 for the 2026-2027 session.
Do these solutions cover every optimization problem in Exercise 2.7?
Yes, all applied word problems and geometric optimization questions from Exercise 2.7 are included, solved step by step with clear constraint equations and second derivative validations.
Are the solutions step-by-step?
Yes. Each problem is solved using the complete 7-step optimization model: understanding the problem, defining variables, writing constraints, formulating a single-variable objective function, finding critical points, verifying extrema using f''(x), and interpreting the physical solution.
Can students read these solutions online?
Yes, the full Exercise 2.7 solution set is available to read online on this page without any download required.
Are these solutions based on the latest PCTB / PECTAA syllabus?
Yes, they are aligned with the current PCTB and PECTAA Mathematics syllabus for the 2026-2027 academic session for Class 12.
Are these solutions free for students?
Yes, Exercise 2.7 solutions on this page are completely free to read for all Class 12 students preparing for Lahore Board, Rawalpindi Board, Gujranwala Board, or any other Punjab Board examinations.