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For all twice differentiable functions f : R $$ \to $$ R, with f(0) = f(1) = f'(0) = 0

JEE · Math · previous-year question

  1. A.f''(x) $$ \ne $$ 0, at every point x $$ \in $$ (0, 1)
  2. B.f''(x) = 0, for some x $$ \in $$ (0, 1)correct
  3. C.f''(0) = 0
  4. D.f''(x) = 0, at every point x $$ \in $$ (0, 1)

Answer

B. f''(x) = 0, for some x $$ \in $$ (0, 1)

Explanation

f : R $$ \to $$ R, with f(0) = f(1) = 0 and f'(0) = 0 $$ \because $$ f(x) is differentiable and continuous and f(0) = f(1) = 0 Applying Rolle’s theorem in [0, 1] for function f(x) f'(c) = 0, c $$ \in $$ (0, 1) Now again $$ \because $$ f'(c) = 0, f'(0) = 0 again applying Rolles theorem in [0, c] for function f'(x) f''(c1) = 0 for some c1 $$ \in $$ (0, c) $$ \in $$ (0, 1)

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