Let `f(x)={((x-1)^2 sin(1/(x-1))-|x|,; x != 1), (-1,; x=1):}` then which one of the following is true?A. f is differential at x=0 but not at x=1B. f is differentiable at x=1 but not at x=0C. f is neither differentiable at x=0 nor at x=1D. f is differentiable at x=0 and at x=1

Let `f(x)={((x-1)^2 sin(1/(x-1))-|x|,; x != 1), (-1,; x=1):}` then whi

1.	Let `f(x)={((x-1)^2 sin(1/(x-1))-\|x\|,; x != 1), (-1,; x=1):}` then which one of the following is true?A. f is differential at x=0 but not at x=1B. f is differentiable at x=1 but not at x=0C. f is neither differentiable at x=0 nor at x=1D. f is differentiable at x=0 and at x=1
Answer» Correct Answer - A We observe that `underset(x to 1)lim (f(x)-f(1))/(x-1)` `underset(x to 1)lim ((x-1)sin ((1)/(x-1)))/(x-1)=underset(x to 1)lim sin ((1)/(x-1))` =An oscillating number between -1 and 1. `therefore underset(x to 1)lim (f(x)-f(0))/(x-0)` `underset(x to 1)lim ((x-1)sin ((1)/(x-1))-sin1)/(x)` `underset(x to 1)lim (2sin(x)/(2(x-1))cos{(2-x)/(2(x-1))})/({(x)/(2(x-1))}2(x-1))` `=-sin 1+cos 1` `Rightarrow` f(x) is differentiable at x=0.