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I came across an expression in 3 variables involving the max and min functions:

$$x = y - \max(x,z) + \min(x,y,z).$$ I need to solve this expression for $y$. Is there any way we can do this? Reading somewhat related answers here such as this and this led me to try splitting this into several cases, but I got lost with both max() and min() functions and three variables. Can anyone give me some insight on the problem (or better yet, a more general method for this procedure?)

Thanks.

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1 Answers1

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First, separate $3$ cases:

  • $\min(x,y,z)=x$
  • $\min(x,y,z)=y$
  • $\min(x,y,z)=z$

In the first case, it is obvious that $\max(x,z)=z$, and in the third, $\max(x,z)=x$. Only in the second case, you have to separate the cases $\max(x,z)=x$ and $\max(x,z)=z$ separately.

5xum
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  • Since my concept on such separation is not clear so I would lie to learn where I am wrong. The outcomes of $\min(x,y,z)$ are the following $$\min (x,y,z)=x, \text{when} \ x\leq \min (y,z)\ \min (x,y,z)=y, \text{when} \ y\leq \min (x,z)\ \min(x,y,z)=z, \text{when} \ z\leq \min (x,y)$$. But what is the outcome of $\min(x,y,z)$ if $x\leq y\geq z$ or $x\geq y\leq z$? – vbm Apr 22 '18 at 20:09