A function f(x) is defined in the interval [1,4) as follows `f(x)={{:(,log_(e)[x],1 le x lt 3),(,|log_(e)x|,3 le x lt 4):}`. Then, the curve y=f(x)

Options

A. is broken at two points
B. is broken at exactly one point
C. does not have a definite tangent at two points
D. does not have a definite tangent at more than two points

Answer

A, C

Explanation

We have
`f(x)={{:(,log_(e)[x],1 le x lt 3),(,|log_(e)x|,3 le x lt 4):}`
`Rightarrow f(x)={{:(,0,1 le x lt 2),(,log_(e)2,2 le x lt 3),(,|log_(e)x|,3 le x lt 4):}`
Clearly, f(x) is everywhere continuous and differentiable except possibility at x=2,3
We observe that `underset(x to 2^(-))lim f(x)=underset(x to2^(-))lim 0=0`
`and underset(x to 2^(+))lim f(x)=underset(x to2^(+))lim log_(e)2=log_(e)2`
Clearly, f(x) is not continuous at x=2
It can be easily seen that f(x) is not continuous at x=3
Hence, f(x) is neither continuous nor differenitable at x=2,3
Thus, the curve y=f(x) is broken at two points and it does not have a definite tangent at these points