Consider a function f(|x| - 1) = |x|, where -1 <= x <= 1. The value of x at which the function attains a maximum, and the maximum value of the function are: A. 0, -1 B. -1, 0 C. 0, 1 D. -1, 2

GATE 2015 · Engineering Mathematics · Maxima Minima · medium

Answer: C. 0, 1

1. Express f explicitly: From f(|x|-1) = |x|: let t = |x|-1, then |x| = t+1, so f(t) = t + 1. For x in [-1,1]: |x| in [0,1], so t = |x|-1 in [-1, 0]. The domain of f is [-1, 0].
2. Find maximum of f on its domain: f(t) = t + 1 is increasing. Maximum at t = 0: f(0) = 0 + 1 = 1. The argument at which the maximum is attained is t = 0.
3. Match with answer options: Option C states x = 0 (argument of f) and max value = 1. This matches: f(0) = 1. Answer is C.