Sol. If possible, let there be a value of λ for which the equation x 3 –3x + λ = 0 has two distinct roots α and β lying between 0 and 1 such that α < β .

Let f(x) = x 3 – 3x + λ .

Since α and β are roots of f(x) = 0. So, by the algebraic interpretation of Rolle's theorem, the equation f ′ (x) = 0 must have a root γ (say) between α and β . But,

f ′ (x) = 3x 2 – 3

∴ f ′ (x) = 0

⇒ 3x 2 –3 = 0 ⇒ x = ± 1

We have, 0 < α < β < 1. Therefore, none of the roots of f ′ (x) = 0 lie between α and β .

This is a contradiction to the fact that f ′ (x) = 0 must have a root between any two roots of f(x) = 0.

Thus, our supposition is wrong. Hence, there is no value of λ for which the equation

x 3 –3x + λ = 0 has two distinct roots lying between 0 and 1.