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BITSAT2024MathematicsContinuity and DifferentiabilityActual

If f(x)= array cc x^2 ( x) (1+x) , & x 0 0, & x=0 array . , then at x=0, f(x) is

Options

  1. Anot continuous
  2. Bcontinuous but not differentiable
  3. Cdifferentiable
  4. Dnot continuous, but differentiable

Correct answer

C. differentiable

Step-by-step solution

_ x 0 f(x)= _ x 0 x^2 ( x) (1+x) = _ x 0 - x ( x) (1+x) x = _ x 0 x ( x)=0 1=0 _ x 0 f(x)=f(0) f(x) is continuous at x=0 Now, _ x 0 f(x+h)-f(x) h _ h 0 f(h)-f(0) h = _ h 0 h^2 h-0 (1+h) _ h 0 h^2 ( h) 1 =0 f(x) is differentiable at x=0

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