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MHT CET202522 Apr 2025Evening ShiftMathematicsApplication of DerivativesActual

The function f (x)= ^4 x+ ^4 x increases if

Options

  1. A0 < x < 8
  2. B4 < x < 2
  3. C3 8 < x < 5 8
  4. D5 8 < x < 3 4

Correct answer

B. 4 < x < 2

Step-by-step solution

Function simplification: Begin by expressing ^4 x + ^4 x as ( ^2 x + ^2 x)^2 - 2 ^2 x ^2 x = 1 - 2 ^2 x ^2 x . Using the identity 2 x x = (2x) , we write 2 ^2 x ^2 x = 1 2 ^2(2x) , yielding f(x) = 1 - 1 2 ^2(2x) . Differentiation: The derivative is f'(x) = - 1 2 2 (2x) 2 (2x) = -2 (2x) (2x) . Apply the identity 2 (2x) (2x) = (4x) to obtain f'(x) = - (4x) . Increasing condition: The function increases where f'(x) > 0 , which requires - (4x) > 0 or (4x) The sine function is negative when its argument is in the interv

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