Explanation

(b) Direction of magnetic field at every point on axis of a current carrying coil remains same though magnitude varies. Hence magnetic induction for whole of the $x$-axis will remain positive
Therefore, (c) and (d) are wrong
Magnitude of magnetic field will vary will $x$ according to the formula, $B = \frac{\mu_{0}NIR^{2}}{2\left( R^{2} + x^{2} \right)^{3/2}}$
Hence, at $x = 0,\ B = \frac{\mu_{0}NI}{2R}$
and when $x \rightarrow \infty,\ B \rightarrow 0$
Slope of the graph will be
$\frac{dB}{dx} = - \frac{3\mu_{0}NIR^{2}.x}{2\left( R^{2} + x^{2} \right)^{5/2}}$
It means, at $x = 0$, slope is equal to zero or tangent to the graph at $x = 0$, must be parallel to $x$-axis.
Hence (b) is correct and (a) is wrong