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AP EAMCET202120 Aug 2021Morning ShiftMathematicsContinuity and DifferentiabilityActual

If f ( x ) , defined below, is continuous at x = 4 , then f x = x 2 + a x + b , 0 ≤ x < 2 3 x + 2 , 2 ≤ x ≤ 4 2 a x + 5 b , 4 < x ≤ 8

Options

  1. Aa = 0    &    b = 0
  2. Ba = 1    &    b = 1
  3. Ca = - 1 & b = 1
  4. Da = 11    &    b = - 18

Correct answer

D. a = 11    &    b = - 18

Step-by-step solution

f x = x 2 + a x + b ,   0 ≤ x < 2 3 x + 2 ,   2 ≤ x ≤ 4 2 a x + 5 b ,   4 < x ≤ 8 If function is continuous at x = 2 and x = 4 , then L . H . L = f x = R . H . L lim x → 2 - f x = f 2 = lim x → 2 + f x 4 + 2 a + b = 8 ⇒ 2 a + b = 4 . . . . . . . . . i lim x → 4 - f x = f 4 = lim x → 4 + f x 14 = 8 a + 5 b . . . . . . . . . i i Solve i   &   i i by elimination,we get a = 11 , b = - 18

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