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NDA2016MathematicsContinuity and DifferentiabilityActual

A function f ( x ) is defined as follows: f(x)= array ccc x+ & for & x [- , 0) x & for & x [0, 2 ] (x- 2 )^2 & for & x ( 2 , ] array . Consider the following statements: 1. The function f ( x ) is continuous at x =0 . 2. The function f(x) is continuous at x= 2 . Which of the above statements is/are correct?

Options

  1. A1 only
  2. B2 only
  3. CBoth 1 and 2
  4. DNeither 1 nor 2

Correct answer

C. Both 1 and 2

Step-by-step solution

Given f(x)= array cc (x+ ) & for x [- , 0) x & for x [0, 2 ] (x- 2 )^2 & for x ( 2 , ] array . For continuity, aligned & f(a)= L.H.L. = R.H.L. & aligned & At x=0 & f(0)= 0= & L.H.L. = _ x 0⁻ f(x-h) &= _ h 0 f(-h) &= _ h 0 (-h+ )= aligned aligned aligned R.H.L. & = _ x 0⁺ f(x+h) & = _ h 0 f(0+h) & = _ h 0 h & = 0= aligned f(0)= L.H.L. = R.H.L. Hence function is continuous at x=0 . Statement (1) is correct. At x= 2 aligned L.H.L. & = _ x ⁻ 2 f(x-h) & = _ h 0 f ( 2 -h ) & = _ h 0 ( 2 -h ) & = 2 =0 aligned aligned RH.L

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