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Manipal MET2012MathematicsDifferentiation

Let f be twice differentiable function satisfying f(1)=1, f(2)=4, f(3)=9 , then

Options

  1. Af^ (x)=2, x R
  2. Bf^ (x)=5=f^ (x) . For some x (1,3)
  3. Cthere exists atleast one x (1,3) such that f^ (x)=2
  4. DNone of the above

Correct answer

C. there exists atleast one x (1,3) such that f^ (x)=2

Step-by-step solution

Let g(x)=f(x)-x^2 g(x) has atleast 3 real roots which are x=1,2,3 (by mean value theorem) g^ (x) has atleast 2 real roots in x (1,3) g^ (x) has atleast 1 real root in x (1,3) f^ (x)=2 for atleast one x (1,3)

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