Answer
a) $\sqrt x$
b) only one curve
Work Step by Step
a)
$f'(x)^{2}$ $=$ $\frac{1}{4x}$
$f'(x)$ $=$ $\frac{1}{2\sqrt x}$
$f(x)$ $=$ ${\sqrt x}$$+$$C$
as we know that curve start from $(1,1)$ point so this point lies on the curve.
$1$$=$$1$$+$$C$
$C$$=$$0$
$f(x)$$=$$\sqrt x$
b) according to given condition we get only one value of arbitrary constant $C$ .so there is only one curve.
