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MHT CET202522 Apr 2025Evening ShiftMathematicsApplication of DerivativesActual

Let f be a function which is continuous and differentiable for all x . If f (1)=1 and f ^ ( x ) 5 for all x in [1,5] , then the maximum value of f(5) is

Options

  1. A5
  2. B20
  3. C6
  4. D21

Correct answer

D. 21

Step-by-step solution

Lagrange's Mean Value Theorem ensures there exists c (1, 5) satisfying f'(c) = f(5) - f(1) 5 - 1 . Given f(1) = 1 , this simplifies to f'(c) = f(5) - 1 4 . Since f'(x) 5 for all x [1, 5] , and c lies within this interval, f'(c) 5 . Substituting gives f(5) - 1 4 5 . Multiplying both sides by 4: f(5) - 1 20 . Adding 1 yields f(5) 21 , establishing the maximum possible value. 21

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