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COMEDK2023Morning ShiftMathematicsContinuity and DifferentiabilityActual

If f(x) = 2 x & ; & - x - 2 a x + b & ; & - 2 < x < 2 x & ; & 2 x . and it is continuous on [- , ] , then

Options

  1. Aa=-1 and b=-1
  2. Ba=1 and b=-1
  3. Ca=-1 and b=1
  4. Da=1 and b=1

Correct answer

B. a=1 and b=-1

Step-by-step solution

For f(x) to be continuous on [- , ] , it must be continuous at x = - 2 and x = 2 . At x = - 2 , the left-hand limit equals the right-hand limit: _ x - /2⁻ 2 x = _ x - /2⁺ (a x + b) 2 (- /2) = a (- /2) + b -2 = -a + b --- (1) At x = 2 , the left-hand limit equals the right-hand limit: _ x /2⁻ (a x + b) = _ x /2⁺ x a ( /2) + b = ( /2) a + b = 0 --- (2) Adding equations (1) and (2): (-a + b) + (a + b) = -2 + 0 2b = -2 b = -1 Substituting b = -1 into equation (2): a - 1 = 0 a = 1 . Answer: a=1 and b=-1

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