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COMEDK2019MathematicsContinuity and Differentiability

aligned f(x) &=2 a-x in -a < x < a &=3 x-2 a in a x aligned Then which of the following is true?

Options

  1. Af(x) is discontinuous at x a
  2. Bf(x) is not differentiable at x
  3. Cf(x) is differentiable at all x a
  4. Df(x) is contimuous at all x < a

Correct answer

D. f(x) is contimuous at all x < a

Step-by-step solution

We have, At x=a aligned & LHL = _ x a (2 a-x)=2 a-a=a & RHL = _ x a (3 x-2 a)=3 a-2 a=a &f(a)=3 a-2 a=a & LHL = RHL =f(a) . aligned So, f(x) is continuous at x=a . Now, when x a, f(x)=3 x-2 a , which is continuous for all x>a . So, f(x) is continuous for all x . Now, at x=a aligned LHD &= _ x a (2 a-x)-a x-a &= _ x a -(x-a) (x-a) =-1 and RHD &= _ x a (3 x-2 a)-a x-a &= _ x a 3(x-a) x-a =3 LHD & RHD aligned So, f(x) is not differentiable at x=a .

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