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The function \mathrm{f(x)=\frac{\tan |\pi[x-\pi]|}{1+[x]^2}} where [x] denotes the greatest integer less than or equal to x, is

Option: 1

discontinuous at some x.


Option: 2

continuous at all x, but f '(x) does not exist for some x.


Option: 3

f '(x) exists for all x, but f "(x) does not exist.


Option: 4

f '(x) exists for all x.


Answers (1)

best_answer

Since  \mathrm{[x-\pi]}  is an integer for all x, therefore  \mathrm{\pi[x-\pi]}  is an integral multiple of \mathrm{\pi} for all x. Hence

tan  \mathrm{|\pi| x-\pi||=0}  for all x. Also  \mathrm{1+[x]^2=0} for all x. Therefore f(x)=0 for all x.
Hence, f(x) is continuous and derivable at all x.

Posted by

manish painkra

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