Statement-1: `If a, b, c in R and 2a + 3b + 6c = 0`, then the equation `ax^(2) + bx + c = 0` has at least one real root in (0, 1).
Statement-2: If f(x) is a polynomial which assumes both positive and negative values, then it has at least one real root.

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

B

Explanation

Let `f(x) = (a)/(3) x^(3) + (b)/(2) x^(2) + cx`.
Clearly, `f(0) = 0 and f(1) = 1//6 (2a + 3b + 6c) = 0`
By Rolle's theorem f'(x) = 0 i.e. `ax^(2) + bx + c = 0` has a real root in (0, 1).
So, statement-1 is true.
Statement-2 is also true but it is not a correct explanation for Statement-1.