Deriving an Expression for an Electric Field along the Z axis

[Cactus]

Homework Statement

Hey, so I've been given this advanced question as part of my physics assignment, and I've been having trouble trying to solve it, so far, I've found multiple answers that all contradict eachother and all of which don't seem right so I'm struggling to find the correct solution. I've attached my current solution below but I still feel this is wrong as the graph doesn't match what I'd expect to happen

The question is as follows:
Derive an expression for the electric field along the vertical line that passes through the negativecharge. How many times does the electric field strength pass through zero? For each such zeropoint of the field, what would happen if a positive test charge was released from rest near to(but not directly on top of) the point where the field is zero?

The question refers to the charge setup below, where one I believe would intuitively expect to find two zero points (One below all the charges, and one above all the charges).

Relevant Equations

Coulombs Law

 

[mjc123]

Your answer looks correct. The field is discontinuous at the negative charge; it tends to positive infinity as you approach the negative charge from below, and negative infinity as you approach it from above. So you follow the black curve for z<d, and the red curve for z>d.

 

[Cactus]

Your answer looks correct. The field is discontinuous at the negative charge; it tends to positive infinity as you approach the negative charge from below, and negative infinity as you approach it from above. So you follow the black curve for z<d, and the red curve for z>d.

Yeah that's mostly the part that I was confused with, how it approached positive infinity even though its approaching a negative charge, however in writing this I think I figured it out. It approaches positive infinity because in this case I've described positive to be "up" along the z axis (So as it approaches the negative charge from below the field should climb to infinity, and be positive as it is traveling up the axis). Thus this would also be the opposite for approaching the charge from above as the field is now down which is negative?

Edit: Also i forgot to say but thanks for pointing this out, I had a friend who also tried the question but set his coord axis on the negative charge rather than in between the positive charges. That confused me at first but plotting his and my graphs together we find we have the exact same graphs except shifted by a value of d, which makes sense cause our coord system starts a distance of d away from each other. But, yeah, thanks for pointing that out cause I was going crazy trying to figure out why my formula wasn't working and why I kept getting the same formula and it turns out it was all because I was misreading the graph.