Statement-1: `If alpha and beta` are real roots of the quadratic equations `ax^(2) + bx + c = 0 and -ax^(2) + bx + c = 0`, then `(a)/(2) x^(2) + bx + c = 0` has a real root between `alpha and beta`
Statement-2: If f(x) is a real polynomial and `x_(1), x_(2) in R` such that `f(x_(1)) f_(x_(2)) lt 0`, then f(x) = 0 has at leat one real root between `x_(1) and x_(2)`.

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

A

Explanation

Let `phi (x) = (a)/(2)x^(2) + bx + c`. Then , `phi(alpha) = (a)/(2) alpha^(2) + b alpha + c`
`rArrphi(alpha)=(a)/(2)alpha^(2)-aalpha^(2)=-(a)/(2)alpha^(2)[{:(becausealpha" is a root of "ax^(2)+bx+c=0),(" "thereforea alpha^(2)+b alpha+c=0):}]`
`and, phi(beta)=(a)/(2)beta^(2)+b beta+c`
`rArr phi(beta)=(a)/(2)beta^(2)+abeta^(2)=(3)/(2)a beta^(2)[{:(becausebeta" is a root of "-ax^(2)+bx+c=0),(" "therefore a beta^(2)+b beta+c=0):}]`
`therefore phi(alpha) phi(beta)=-(3a^(2))/(4)(alpha beta)^(2) lt 0`
So, `phi(x)=0` has a real root between `alpha` and `beta`.
Hence, statement-1 is correct and statement-2 is a correct explanation for statement-1.