If α and β (α < β) be two different real roots of the equation ax2 ...
Also, f(x) is continuous and differentiable in [] as it is a polynomial function or x.
Hence, by Rolle's theorem, there exists a k in (), such that
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If α and β (α < β) be two different real roots of the equation ax2 ...
Explanation:
Given:
α and β are two different real roots of the equation ax^2 + bx + c = 0, where α < />
Proof:
1. Relationship between roots and coefficients:
For a quadratic equation ax^2 + bx + c = 0, the sum of the roots is given by -b/a and the product of the roots is given by c/a. Since α and β are the roots of the equation, we have α + β = -b/a and αβ = c/a.
2. Relationship between roots and average of roots:
The average of the roots of a quadratic equation is given by (α + β)/2. Since α < β,="" the="" average="" of="" the="" roots="" is="" closer="" to="" α="" than="" to="" />
3. Average of roots and coefficient b:
From the relationship between roots and the average of roots, we have (α + β)/2 = -b/2a. Since the average of the roots is closer to α, we can say that α < />
Therefore, the correct statement is:
α < -b/2a="" />< β.="" hence,="" option="" 'c'="" is="" the="" correct="" answer.="" β.="" hence,="" option="" 'c'="" is="" the="" correct="" />