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    				| 1. | 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 . | 
| Answer» 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)` | |