JEE Main 2025 Jan 23 Shift 1, Mathematics Q24: Maxima & Minima

JEE Main2025Jan 23, Shift 1Mathematics

If the set of all values of $a$ , for which the equation $5 x^{3} - 15 x - a = 0$ has three distinct real roots, is the interval $(α, β)$ , then $β - 2 α$ is equal to ________

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Solution

Treat $a = 5 x^{3} - 15 x$ as $a = f (x)$ and ask for which $a$ the horizontal line $y = a$ crosses the cubic at three distinct points.

$f^{'} (x) = 15 (x^{2} - 1) = 0$ at $x = \pm 1$ , giving a local maximum at $x = - 1$ with $f (- 1) = - 5 + 15 = 10$ , and a local minimum at $x = 1$ with $f (1) = 5 - 15 = - 10$ .

Three real roots occur exactly when $a$ lies strictly between the local min and local max values: $a \in (- 10, 10)$ .

So $α = - 10$ , $β = 10$ , and $β - 2 α = 10 - 2 (- 10) = 30$ .

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JEE Main 2025 Jan 23 Shift 1, Mathematics Q24: Maxima & Minima

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