Question

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

Answer 1

f (1/2)=2.......(given)
and f(x) = x² + a, for x ≥ 0

`f(1/2)=lim_(x->(1/2))f(x)=lim_(x->(1/2))x^2+a=1/4+a`

2=1/4+a

a=7/4

Also, f(x) is continuous at x = 0

`lim_(x->0^-)f(x)=lim_(x->0^+)f(x)=0`

`lim_(x->0^-)2sqrt(x^2+1)+b=lim_(x->0^-)x^2+a`

`2sqrt(0+1)+b=0+7/4`

`b=-1/4`

`a=7/4 and b=-1/4`