Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined as $f(x)=|x|+\left|x^{2}-1\right|$. The total number of points at which $f$ attains either a local maximum or a local minimum is

JEE Advanced · 2012 · 08 Apr 2012 · Paper 1 · Application of Derivatives

Let

f : R \to R

be defined as

f (x) = ∣ x ∣ + x^{2} - 1

. The total number of points at which

f

attains either a local maximum or a local minimum is

Answer: 5

f (x) = ∣ x ∣ + ∣ x^{2} - 1∣

. The critical points occur where the derivative changes sign. We need to consider the points where

x = 0

x = \pm 1

(where the expressions inside absolute values change sign). Analyzing the derivative in each interval:

f^{'} (x) = ⎩ ⎨ ⎧ 2 x - 1, - (2 x + 1), 1 - 2 x, 2 x + 1, x < - 1 - 1 < x < 0 0 < x < 1 x > 1

The derivative changes sign at

x = - 1, - \frac{1}{2}, 0, \frac{1}{2}, 1

, giving 5 points of local extrema.

JEE Advanced · 2012 · 08 Apr 2012 · Paper 1 · Integer answer