Let f(x) be a function given by `f(x) =(x^2+1)/([x])`
where [] denotes the greatest interger function .Then f(x) is monotonically

Options

A. increasing on [1,4)
B. decreasing [1,4)
C. increasing on [1,2)
D. decreasing on [2,3)

Answer

C

Explanation

we have
`f(x)=(x^2+1)/[x] for all x In [1,4)`
`rArr f(x){{:(x^2+1"," for all x in [1,2)),((x^2+1)/(2)"," for all x in [2,3)),((x^2+1)/(3)"," for all x in [3,4)):}`
`rArr f(x){{:(2x"," for all x in [1,2)),(x"," for all x in [2,3)),((2x)/(3)"," for all x in [3,4)):}`
Clearly `f'(x) gt 0` in each of the intevals (1,2)(2,3) and (3,4) so f(x) is increasing in each of the increasing on [1,4) because values of f(x) in (15,2) are grater than the value of f(x) in (2,3)