Let f(x)=|x|+1 and g(x)=[x]-1 , where [.] is the greatest integer function. Let h(x)= f(x) g(x) Consider the following statements: 1. f(x) is differentiable for all x 0 2. g(x) is continuous at x=0.0001 3. The derivative of g(x) at x=2.5 is 1 Which of the statements given above are correct?

Step-by-step solution

and aligned f(x) & =|x|+1 g(x) & =[x]-1 h(x) & = f(x) g(x) = |x|+1 [x]-1 aligned (1) For, x 0 aligned & f(x)=-x+1 & f(x)=-1 aligned f(x) is differentiable x 0 Statement (1) is correct. (2) At x=0.0001g(x) is continuous everywhere except for integer value as [ x ] is continuous x R except integers. So, g(x) is continuous at x=0.0001 . Statement (2) is correct. aligned & (3) g(x)=[x]-1 & [x] Always a b integer & aligned g^ (x) & = d d x [x]-0 g^ (x) & =0-0=0 aligned aligned Statement (3) is incorrect. Hence, (1) and