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NDA2020MathematicsContinuity and DifferentiabilityActual

Consider the following statements for f(x)=e^ -|x| : 1. The function is continuous at x=0 . 2. The function is differentiable at x=0 . 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

A. 1 only

Step-by-step solution

(1) f(x)=e^ -|x| , f(0)=e^ -|0| =1 . gathered f(x)=e^ -x , for x 0 =e^x, for x 0 gathered aligned & LHL = _ x 0⁻ f(x)= _ x 0⁻ e^x=1 & RHL = _ x 0⁺ f(x)= _ x 0⁺ e^ -x =1 aligned As, f(0)= LHL = RHL Thus f(x) is continuous at x=0 . (2) LHD = _ x 0⁻ f^ (x)= _ x 0⁻ e^x=e^0=1 RHD = _ x 0⁺ f^ (x)= _ x 0⁺ -e^ -x =-e^0=-1 As LHD RHD Hence, f(x) is not differentiable at x=0 .

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