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Question: The number of values of k for which the equation x<sup>3</sup> – 3x + k = 0 has two different roots...

The number of values of k for which the equation

x3 – 3x + k = 0 has two different roots lying in the interval

(0, 1) are –

A

3

B

2

C

Infinitely many

D

No value of k satisfies the requirement

Answer

No value of k satisfies the requirement

Explanation

Solution

Let ƒ(x) = x3 – 3x + k. Let, if possible, a, b Ī (0, 1) be different roots of ƒ(x) = 0. ƒ(x) is continuous on [0, 1] and differentiable on (0, 1).

Also, ƒ(1) = ƒ(2) = 0

\ By Rolle’s theorem there exists c Ī (0, 1) such that

ƒ¢ (3) = 0 Ž 3c2 – 3 = 0 Ž c = ± 1

\ no value of k satisfies the condition.