Determine the values of x for which the following functions fails to be continuous or differentiable `f(x) = {{:((1-x)",",x lt 1),((1-x)(2-x)",",1 le x le 2),((3-x)",",x gt 2):}` justify your answer.

Options

A. x=1
B. x=2
C. x=1,2
D. none of these

Answer

B

Explanation

We have
`underset(x to 1^(-))lim f(x)=underset(x to 1^(-))lim (1-x)=0`
`underset(x to 1^(+))lim f(x)=underset(x to 1^(+))lim (1-x)(2-x)=0, and f(1)=0`
`therefore underset(x to 1^(-))lim f(x)=underset(x to 1^(+))lim f(x)=f(1)`
So, f(x) is continuous at x=1
Now,
`("LHD at x=1")=((d)/(dx)(1-x))_(x=1)=-1`
`("RHD at x=1")=((d)/(dx)(1-x)(2-x))_(x=1)=-1`
`therefore ("LHD at x=1")=("RHD at x=1")`
So, f(x) is differentiable at x=1
Now,
`underset(x to 2^(-))lim f(x)=underset(x to 2^(-))lim (1-x)(2-x)=0`
and
`underset(x to 2^(+))lim f(x)=underset(x to 2^(+))lim 3-x=1`
`therefore underset(x to 2^(-))lim f(x) ne underset(x to 2^(+))lim f(x)`
Thus, f(x) is discontinuous and non-differentially at x=2. Hence the only point of discontinuity and non-differentibility of `f(x) is x =2`