A long, straight wire of radius r carrying a uniformly distributed...

Let the observer look along the direction of the current, meaning the current is flowing away from the observer into the plane.

According to the right-hand grip rule, the magnetic field lines produced by the straight wire form concentric clockwise circles around the wire.

The external uniform magnetic field is directed vertically upward.

To obtain the maximum resultant magnetic field, the magnetic field produced by the wire must be in the same direction as the external magnetic field (i.e., vertically upward).

Looking at the clockwise magnetic field lines, the tangent to the field lines points vertically upward at the leftmost points of the wire. At the rightmost points, the field points vertically downward.

The magnitude of the magnetic field due to the wire is maximum at its surface and is given by $\dfrac{\mu_0 i}{2\pi r}$.

Thus, at the leftmost points on the wire's surface, both the external magnetic field and the wire's magnetic field are parallel and directed vertically upward. The maximum resultant magnetic field is the sum of their magnitudes:

$B_{net} = B + \dfrac{\mu_0 i}{2\pi r}$