Consider a quadratic equation a x^2+b x+c=0 , where 2 a+3 b+6 c=0 and let g(x)=a x^3 3 +b x^2 2 +c x . Statement 1: The quadratic equation has at least one root in the interval (0,1) . Statement 2: The Rolle's theorem is applicable to function g(x) on the interval [0,1] .

Step-by-step solution

Let g(x)= a x^3 3 +b x^2 2 +c x g ^ (x)=a x^2+b x+c Given: a x^2+b x+c=0 and 2 a+3 b+6 c=0 Statement-2: (i) g(0)=0 and g(1) aligned & = a 3 + b 2 +c= 2 a+3 b+6 c 6 & = 0 6 =0 & g(0)=g(1) aligned (ii) g is continuous on [0,1] and differentiable on (0,1) By Rolle's theorem k (0,1) such that g^ (k)=0 This holds the statement 2. Also, from statement-2, we can say a x^2+b x+c=0 has at least one root in (0,1) . Thus statement-1 and 2 both are true and statement-2 is a correct explanation for statement-1.