Question
For the following two (02) items: Consider the function f(x) = 1 - [3] (x-1)^2 . The function has
NDA
Mathematics
Application of Derivatives
2025
NDA 2025 (Phase 2)
NDA Mathematics Question (2025) — Solution
Options
- A. a minimum at x = 1
- B. a maximum at x = 1
- C. neither maximum nor minimum at x = 1
- D. no extremum
Answer
B. a maximum at x = 1
Step-by-step solution
Given f(x) = 1 - (x-1)^ 2/3 Differentiating with respect to x: f'(x) = - 2 3 (x-1)^ -1/3 = -2 3(x-1)^ 1/3 For x 0. Thus, f(x) is strictly increasing. For x > 1, x-1 > 0 f'(x) Since f'(x) changes sign from positive to negative as x passes through 1, f(x) has a local maximum at x = 1. Alternatively, (x-1)^ 2/3 0 for all real x. 1 - (x-1)^ 2/3 1 f(x) f(1) for all x. Hence, f(x) has a maximum at x = 1. Answer: a maximum at x = 1
Related: Mathematics — Application of Derivatives · All PYQ Banks