Lagrange multipliers
When a (continuous) function of one variable is strictly crescent or decrescent we don't have maximum or minimum points unless we set a closed interval, in which case the boundaries themselves are going to be the maximum and minimum points. For functions of two variables we can do the same and set a subdomain to limit our search for maximum and minimum points. The difference is that domain of a two variable function lies in [math]\displaystyle{ \mathbb{R}^2 }[/math], which means that the subdomain is going to be all points from a certain subset, a circumference for example. For functions of three variables we can't see the graph, but we can plot level surfaces and visualise constrains in [math]\displaystyle{ \mathbb{R}^3 }[/math].
Many textbooks begin the explanation of Lagrange's multipliers with the partial derivatives and a system of equations. I'm going to use the graphical interpretation first to make it easier to understand the concept:
The function is [math]\displaystyle{ f(x,y) = x + y }[/math] and the constrain is [math]\displaystyle{ x^2 + y^2 = 1 }[/math]. As you can see, the function's domain has been restrained to all points that belong to the equation of a circumference with radius equal to 1. If we displace the circumference along the z axis, [math]\displaystyle{ f(x,y) = z }[/math] in this case, we are going to have it intersect the function's surface at some z height.
