Statement-1: There is a value of k for which the equation `x^(3) - 3x + k = 0` has a root between 0 and 1.
Statement-2: Between any two real roots of a polynomial there is a root of its derivation.

Options

A. Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False.
D. Statement-1 is False, Statement-2 is True.

Answer

D

Explanation

Statement-2 is the algebraic interpretation of Rolle's Theorem. So, it is true.
Let `f(x) = x^(3) - 3x + k`. Let there be a value of k for which f(x) = 0 has a real root between 0 and 1. Then, f'(x) = 0 must have a root between 0 and 1.
We have, `f'(x) = 3x^(2) - 3`
`therefore" "f'(x) = 0 rArr x = +- 1`
Clearly, none of the roots of f'(x) = 0 lies between 0 and 1. So, our supposition is wrong. Hence, there is no value of k for which the equation `x^(3) - 3x + k = 0` has a root between 0 and 1.
So, statement-1 is not true.