JEE Main 2025 Apr 3 Shift 2, Mathematics Q1: Some Important Properties Of Functions

JEE Main2025Apr 3, Shift 2Mathematics

Let $f : R \to R$ be a function defined by $f (x) = ∣∣ x + 2∣ - 2∣ x ∣∣$ . If $m$ is the number of points of local minima and $n$ is the number of points of local maxima of $f$ , then $m + n$ is

A
$5$
B
$3$
C
$2$
D
$4$

Solution

$f (x) = ∣ ∣ x + 2∣ - 2∣ x ∣ ∣$

Breaking into intervals,

$f(x)= \begin{cases} 2-x, & x<-2\\[2mm] -(3x+2), & -2\le x\le-\dfrac23\\[2mm] 3x+2, & -\dfrac23\le x<0\\[2mm] 2-x, & 0\le x\le2\\[2mm] x-2, & x\ge2 \end{cases}$

Hence, local minima occur at $x = - \frac{2}{3}, 2$ and a local maximum occurs at $x = 0$ .

$∴ m = 2, n = 1$

$m + n = 3$

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JEE Main 2025 Apr 3 Shift 2, Mathematics Q1: Some Important Properties Of Functions

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