Let f(x)  be a twice differentiable function for all real values of x and satisfiesf(1)=1,f(2)=4,f(3)=9. then which of the following is definitely true?

Let g(x)=f(x)−x2.  We have g(1)=o,g(2)=0,g(3)=0  [∵f(1)=1,f(2)=4,f(3)=9] From Rolle’s theorem on g(x),g'(x)=0  for at least x∈(1,2). let g'(c1)=0  where c1,∈(1,2). Similarly, g(x)=0  for at least one x∈(2,3).  let g'(c2)=0  where c2∈(1,2). Therefore,g'(c1)=g'(c2)=0 By Rolle’s theorem, at least one x∈(c1,c2) such that g''(x)=0⇒fi'(x)=2  for some x∈(1,3).