Let `f(x)={{:(int_(0)^(x)(5+|1-t|)dt","," if "xgt 2),(5x+1","," if "x le 2):}` then the function is

Options

A. `f(x)` is continuous everywhere
B. `f(x)` is not continuous at `x=2`
C. `f(x)` is not difference at exactly three points
D. `f(x)` is continuous and differentiable at `x=2`

Answer

A

Explanation

for `xlt2 f^(x)=int_(0)^(x){5+|1-t|}dt`
`=int_(0)^(1){5+1-i}d+int_(1)^(x)(5+t-t)dt`
`[6t-(t^(2))/2]_(0)^(1)+[4t-(t^(2))/2]_(0)^(x)=1+4x+(x^(2))/2`
`:.f(x)={(1+3x+(x^(2))/2, xlt2),(5x+1, xge2:}`
`f(2)=11(f^(-2))=9+211,f(2)=11`
`:.f(x)` is continuous at `x=2impliesf(x)` is contiuous
`f^(')(x)={(4+x,xlt2),(5,xge2):}`
`f^(')(2^(-))=6,f^(')(2)=5`
`f(x)` is not differetiable at `x=2`