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AP EAMCET2011MathematicsContinuity and Differentiability

If [x] denotes the greatest integer not exceeding x and if the function f defined by f(x)= cases a+2 x x^2 & , x < 0 b [x+4] & , x 0 cases is continuous at x=0 , then the ordered pair (a, b) is equal to

Options

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

Correct answer

B. (-2,-1)

Step-by-step solution

Given, f(x)= cases a+2 x x^2 & , x < 0 b [x+4] & , x 0 cases At x =0 LHL = _ x 0⁻ f(x)= _ x 0 a+2 x x^2 aligned & = _ x 0 a+2 (1- x^2 2 ! + x^4 4 ! - ) x^2 & = _ x 0 (a+2)+2 (- x^2 x ! + x^4 4 ! - ) x^2 & = _ x 0 0+2 ( -x^2 2 ! + x^4 4 ! - ) x^2 =-1 & [ f(x) is continuous so we take a+2=0 ] & RHL = _ x 0⁺ f(x)= _ x 0 b [x+4] & =b 4 =b & But LHL = RHL & -1=b and a=-2 & aligned

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