# what does $\min()$ and $\max()$ mean in a function?

Could someone please explain to me what is the difference between these equations how it would affect the result and the way they are solved?

$$Q2(K,L) = A\min(BK,CL)\text{ s.t. }A,B,C > 0$$ $$Q2(K,L) = A\max(BK,CL)\text{ s.t. }A,B,C > 0$$ $$Q2(K,L) = BK,CL\text{ s.t. }A,B,C > 0$$

(Do note that invented the last two equations so I'm not sure if they make any sense mathematically)

I'm encountering these equations quite often in economics and never really understood how to solve them.

Thank you very much for your help!

Usually $$\min(x,y)$$ denotes the smallest of $$x$$ and $$y$$, and $$\max(x,y)$$ the greatest.

• Thank you very much for your answer Bernard! How would this influence the way we would solve this equation? if you could maybe show some kind of example that would be amazing (with a different equation if it's easier) Mar 30 '19 at 22:18
• It does not seem to be an equation, but the definition of a function of the two variables $K$ and $L$, as far as I understand, not knowing the context. Mar 30 '19 at 22:20

Here is a possible interpretation of such a function.

$$Q(B,S,N) = \min(\frac1{2}B,\frac1{6}S, \frac1{12}N)\text{ s.t. }B,S,N, Q \in \mathbb N_0$$

This function can be seen as production function of a chair. You probably need $$2$$ boards ($$B$$) for the back rest and the seat surface. $$6$$ wooden sticks ($$S$$): 4 for the chair legs and 2 to fix the back rest. And probably $$12$$ nails (N) to fix all the parts

Now with Q you can evaluate how many chairs you can produce if some quantities of boards, wooden sticks and nails are available. Let´s say $$B=30, S=100, N=190$$ Then the output of the production is

$$\min\left(\frac1{2}\cdot 30,\frac1{6}\cdot 100, \frac1{12}\cdot 190\right)=\min\left(15,16\frac2{3}, 15\frac5{6}\right)=15 \ \ \color{grey}{\textrm{chairs}}$$

• though how does the $min()$ change the answer? Mar 31 '19 at 12:33
• @Fozoro The min function selects the lowest number of the three numbers since for every chair certain numbers of boards, wooden sticks and nails are needed. In this example the number of chairs is a positive integer. That means that you have to round off the numbers. Mar 31 '19 at 13:26
• so basically it tells us to solve the function with the smallest numbers possible, is that correct? Mar 31 '19 at 13:47
• @Fozoro Yes, if there are no other restrictions then $\min(3.42, 5.2, 4.1)=3.42$ Mar 31 '19 at 14:00