Let `f(x)={{:(3x^(2)-4sqrtx+1,"for",xlt1),(ax+b,"for",xge1):}`
If f(x) is continuous and differentiable for all real values in its domain then

Options

A. `a=b=4`
B. `a=b=-4`
C. `a=4 and b=-4`
D. `a=-4 and b=4`

Answer

C

Explanation

`f(x)={{:(3x^(2)-4sqrtx+1,"for",xlt1),(ax+b,"for",xge1):}`
for continuity at `x=1, f(1^(+))=f(1^(-))`
`rArr" "a+b=3-4+1`
`rArr" "a+b=0`
for differentiability at x = 1,
`f'(1^(+))=f'(1^(-))`
`rArr" "a=6-4=4`
`rArr" "b'=-4`