Question
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 (x) = x3 – 3x + k. Let, if possible, a, b Ī (0, 1) be different roots of (x) = 0. (x) is continuous on [0, 1] and differentiable on (0, 1).
Also, (1) = (2) = 0
\ By Rolle’s theorem there exists c Ī (0, 1) such that
¢ (3) = 0 Ž 3c2 – 3 = 0 Ž c = ± 1
\ no value of k satisfies the condition.