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MHT CET202527 Apr 2025Evening ShiftMathematicsContinuity and DifferentiabilityActual

If f (x)= cases -2 x & , x - 2 a x+ b & , - 2 < x < 2 x & , x 2 cases is continuous at x = - 2 and x= 2 , then the value of 2 a+ b is

Options

  1. A(0, -13 5 , 2 5 )
  2. B(0, 13 5 , 2 5 )
  3. C(0, 13 5 , 2 )
  4. D(0, -13 5 ,-2 )

Correct answer

A. (0, -13 5 , 2 5 )

Step-by-step solution

For f(x) to be continuous at x = - 2 and x = 2 , the left and right limits must equal the function values at these points. At x = - 2 , the left limit is 2 and the right limit is -a + b , yielding -a + b = 2 . At x = 2 , the left limit is a + b and the right limit is 0 , giving a + b = 0 . Solving this system, 2b = 2 so b = 1 , and a = -1 . Thus 2a + b = -2 + 1 = -1 . Final answer: -1

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