JEE Main 2025 Apr 4 Shift 2, Mathematics Q2: Maxima & Minima

JEE Main2025Apr 4, Shift 2Mathematics

Let $f$ be a differentiable function on $R$ such that $f (2) = 1$ , $f^{'} (2) = 4$ . Let $x \to 0 lim (f (2 + x))^{3/ x} = e^{α}$ . Then the number of times the curve $y = 4 x^{3} - 4 x^{2} - 4 (α - 7) x - α$ meets $x$ -axis is :-

Solution

The limit has the indeterminate form $1^{\infty}$ . Take logarithms:

$x \to 0 lim \frac{3}{x} ln f (2 + x) = x \to 0 lim \frac{3 ( f ( 2 + x ) - 1 )}{x} = 3 f^{'} (2) = 12$ .

So $α = 12$ and the curve becomes $y = 4 x^{3} - 4 x^{2} - 20 x - 12 = 4 (x^{3} - x^{2} - 5 x - 3)$ .

Factor: $x^{3} - x^{2} - 5 x - 3 = (x - 3) (x + 1)^{2}$ . So roots are $x = 3$ and a double root at $x = - 1$ .

The curve touches the $x$ -axis at $x = - 1$ (one distinct intersection) and crosses at $x = 3$ (another). Hence it meets the $x$ -axis at $2$ distinct points.

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JEE Main 2025 Apr 4 Shift 2, Mathematics Q2: Maxima & Minima

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