If f(x)`=x^(2)+a`, for `xge0`, f(x)`=2sqrt(x^(2)+1)+b,"for"xlt0andf((1)/(2))=2`, is continuous at x = 0 , find a and b .

Explanation

Given, `f(x)={{:(x^(2)+a",","for"xge0),(2sqrt(x^(2)+1)+b",","for"xlt0):}`
Now , `f((1)/(2))=((1)/(2))^(2)+a`
According to question,
`(1)/(4)+a=2`
`a=2-(1)/(4)`
`a=(7)/(4)`
f is continuous at x =0.
R.H.L.
`lim_(xto0^(+))f(x)=lim_(xto0^(+))(x^(2)+a)`
`=0^(2)+a`
=a
L.H.L.
`lim_(xto0^(-))f(x)=lim_(xto0^(-))2sqrt(x^(2)+1)+b`
=2 +b
According to question,
R.H.L. = L.H.L.
a=2+b
But `a=(7)/(4)`
`(7)/(4)=2+b`
`(7)/(4)-2=b`
`b=-(1)/(4)`