If I remember correctly, Cate and Nelson (1971) use a sum of squares approach to determine how best break the data into two groups.

I wrote a function in R to do a Cate–Nelson analysis.   The most recent version of this function is found at Mangiafico (2015).

For coding for SAS, and a description of using Cate–Nelson analysis in areas other than soil-testing see Mangiafico (2013).  This paper also has a general description of how to do the analysis.

References

Mangiafico, 2015. Cate–Nelson Analysis. In An R Companion for the Handbook of Biological Statistics. http://rcompanion.org/rcompanion/h_02.html

Mangiafico, S.S. 2013. Cate-Nelson Analysis for Bivariate Data Using R-project. J.of Extension 51:5, 5TOT1. http://www.joe.org/joe/2013october/tt1.php

Cate, R. B., & Nelson, L.A. (1971). A simple statistical procedure for partitioning soil test correlation data into two classes. Soil Science Society of America Proceedings 35, 658–660.

I cannot provide the method but can advise as follows.  First, the test values must  saturate the yield response.  The preferred method is to fit the data with a continuous, asymptotic function and estimate the critical limit as the soil test value corresponding with 90% maximum yield,

Unfortunately, the Cate and Nelson (1971) is behind a pay wall here:  https://www.soils.org/publications/sssaj/abstracts/35/4/SS0350040658

(Don't get me started on that topic!)

If I remember correctly, they used an iterative process.  Order the data by x values.  If you have 10 data points, assign 1 to group A, and 9 to group B, and then calculate the Sum of Squares, as if it were the Explained SS or Model SS from an ANOVA. (Or something similar using SS).  You record this SS.  Then you do this again with 2 in group A and 8 in group B.  Do this for all possible critical-x values, and choose the one with the highest SS.  To get critical-y, you choose the y value that will minimize the sum of the points that fall into quadrants I and III.

This is the process that my function follows.  If you can use R, I recommend trying my function.  It's pretty good.

Also, I recommend looking at some of the references listed in Mangiafico (2013) if you can a hold of any.  They will give you good examples of how to write up results.

@Paul Milham,  I think it depends, too, what your data looks like. 

For example, if you have data that look like the attached image01, it makes sense to fit a continuous asymptotic function.

But if your data looks like that in the attached image02, things can get really squirrelly if you try to fit a continuous function.  That's where Cate–Nelson is particularly useful.

References:

2006. S.S. Mangiafico and K. Guillard. Anion exchange membrane soil nitrate predicts turfgrass color and yield.

Crop Science 46:569–577. crop.scijournals.org/cgi/reprint/46/2/569.pdf .

2007. S.S. Mangiafico and K. Guillard. Nitrate leaching from Kentucky bluegrass soil columns predicted with anion exchange membranes. Soil Science Society of America Journal 71:210–224.

soil.scijournals.org/cgi/reprint/71/1/219.pdf .