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

JEE Main · 2025 · 23 Jan 2025 · Shift 1 · Application of Derivatives

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

Answer: 30

Let

f (x) = 5 x^{3} - 15 x

. The equation

f (x) = a

has three distinct real roots exactly when the horizontal line

y = a

cuts the cubic between its local maximum and local minimum.

f^{'} (x) = 15 x^{2} - 15 = 15 (x - 1) (x + 1)

Critical points at

x = \pm 1

f (- 1) = - 5 + 15 = 10 (local max), f (1) = 5 - 15 = - 10 (local min)

Three distinct real roots occur for

a \in (- 10, 10)

, so

α = - 10

β = 10

β - 2 α = 10 - 2 (- 10) = 30

JEE Main · 2025 · 23 Jan 2025 · Shift 1 · Integer answer