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 differentiable at x=0 . 2. The function f ( x ) is differentiable at x = 2 . Which of the above statements is/are correct?

Step-by-step solution

For differentiability, L.H.D. = R.H.D. Thus at x=0 aligned L.H.D. & = _ h 0 f(0-h)-f(0) -h & = _ h 0 f(-h)-f(0) -h & = _ h 0 (-h+ )-( 0) -h =1 L.H.D. = & 1 R.H.D. & = _ h 0 f(0+h)-f(0) h & = _ h 0 f(h)-f(0) h & = _ h 0 h- 0 h & = _ h 0 [ h-1] h aligned aligned & = _ h 0 [1- h^2 2! + h^4 4! -1 ] h & = _ h 0 [1- 1 2 h^2+ 1 24 h^4 . .-1 ] h & = _ h 0 [- 1 2 h^2+ 1 24 h^4 ] h =0 aligned L.H.D. R.H.D. So at x=0 function is not differentiable. Statement (1) is not correct. At x= 2 aligned f ( 2 ) & = 2 =0 R.H.D. & = _ h