JEE Main 2025 Jan 28 Shift 2, Mathematics Q14: Maxima & Minima

JEE Main2025Jan 28, Shift 2Mathematics

Let $f : [0, 3] \to A$ be defined by $f (x) = 2 x^{3} - 15 x^{2} + 36 x + 7$ and $g : [0, \infty) \to B$ be defined by $g (x) = \frac{x ^{2025}}{x ^{2025} + 1}$ . If both the functions are onto and $S = {x \in Z : x \in A or x \in B}$ , then $n (S)$ is equal to :

A
$30$
B
$36$
C
$29$
D
$31$

Solution

Since both are onto, $A$ and $B$ equal the respective ranges.

For $f$ : $f^{'} (x) = 6 x^{2} - 30 x + 36 = 6 (x - 2) (x - 3)$ , so critical points lie at $x = 2, 3$ within $[0, 3]$ . Evaluating: $f (0) = 7$ , $f (2) = 35$ , $f (3) = 34$ . So range $A = [7, 35]$ .

For $g$ : writing $g (x) = 1 - \frac{1}{x ^{2025} + 1}$ , as $x$ varies over $[0, \infty)$ , $g (x)$ traverses $[0, 1)$ . So $B = [0, 1)$ .

Integers in $A \cup B$ : from $B$ only $0$ ; from $A$ the integers $7, 8, \dots, 35$ , which is 29 values. Total $1 + 29 = 30$ .

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JEE Main 2025 Jan 28 Shift 2, Mathematics Q14: Maxima & Minima

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