Exercise 2.4 Class 12 Math Solutions - Unit 2 Further Differentiation, Punjab Board

Step-by-step solutions to Exercise 2.4, Unit 2: Further Differentiation, for Class 12 Punjab Board students. Covers monotonicity (increasing and decreasing intervals), stationary and critical points, relative extrema via the First and Second Derivative Tests, and finding global (absolute) extrema on closed intervals [a, b], worked out question by question.

This page provides complete solved solutions for Exercise 2.4 of Unit 2: Further Differentiation, Class 12 Mathematics, for Punjab Board (PCTB / PECTAA). Every question is solved step by step, covering intervals of increasing and decreasing functions, locating stationary and critical points, classifying relative maxima and minima using the Second Derivative Test, and finding global absolute extrema on closed intervals [a, b].
Back
Exercise 2.3
Next
Exercise 2.5
Advertisement

Exercise Overview

Exercise 2.4 of Class 12 Mathematics represents the analytical pinnacle of single-variable differential calculus under the Punjab Board PCTB / PECTAA curriculum: the determination of function monotonicity and optimization of extrema. The exercise begins by using the sign of the first derivative to classify functions as strictly increasing (f'(x) > 0) or strictly decreasing (f'(x) < 0) over specified domain intervals.

Students then explore critical points (where (f'(x) = 0) or (f'(x)) does not exist) and stationary points (where (f'(x) = 0)). The Second Derivative Test is rigorously applied to classify local extrema: a critical point (c) yields a relative maximum if (f''(c) < 0) (concave downward) and a relative minimum if (f''(c) > 0) (concave upward). Furthermore, the exercise details the Closed Interval Method for finding global (absolute) extrema of continuous functions on [a, b] by evaluating and comparing function values at all interior critical points and domain endpoints.

For Punjab Board HSSC Part 2 examinations across all nine educational boards (Lahore, Rawalpindi, Gujranwala, Faisalabad, Multan, Bahawalpur, Sargodha, Sahiwal, and D.G. Khan), Exercise 2.4 is one of the highest-yield sources of calculus long questions and analytical MCQs. These step-by-step solutions provide clean derivative evaluations, concavity checks, and structured endpoint tables for each problem.

FAQs

Are these Exercise 2.4 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 topics assigned to Unit 2, Exercise 2.4 for the 2026-2027 session.
Do these solutions cover every question in Exercise 2.4?
Yes, all questions from Exercise 2.4 are included, worked through step by step covering monotonicity tests, critical points, local extrema tests, and closed-interval global optimization.
Are the solutions step-by-step?
Yes. Each question details the analytical procedure-computing f'(x) to find critical values, testing signs for increasing/decreasing intervals, applying the Second Derivative Test f''(x) for concavity, and comparing boundary values with critical points for global extrema.
Can students read these solutions online?
Yes, the full Exercise 2.4 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 2.4 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.