Consider the following in respect of the function f(x)= array l 2+x, x 0 2-x, x 0 array . 1. _ x 1 f (x) does not exist. 2. f(x) is differentiable at x=0 3. f(x) is continuous at x=0 Which of the above statements is/are correct?

Step-by-step solution

aligned & For x 1 & _ x 1 f(x)= _ x 1 2+x=2+1=3 & For x 1 & _ x 1⁻ f(x)= _ x 1⁻ 2+x=2+1=3 aligned So, _ x 1 f(x) exist. At x=0 RHL: _ h 0⁺ f(0+h)= _ h 0 2+h=2 LHL _ h 0⁻ f(0-h)= _ h 0 2-h=2f(0)=2+0=2 . So , RHL = LHL =f(0) f(x) is continuous at x=0 Differentiability at x=0 LHD: _ h 0⁻ f(0-h)-f(0) -h = _ h 0⁻ 2+h-2 -h = -h h =-1 RHD: _ h 0⁺ f(0+h)-f(0) h = _ h 0⁺ 2+h-2 h =1 Since LHD RHD So, f(x) is not differentiable at x=0 .