Find minimum or maximum of multi-variable function

[jlind]

(08-28-2019 07:14 AM)teerasak Wrote:  I'm not sure whether there is a function or command which can find the minimum or maximum of multi-variable function e.g.

find min of Sqrt((x-14)^2 + (y-55)^2) + Sqrt((x-23)^2 + (y-95)^2)

Or any application which can help finding this kind of problem?

This is a multivariate calculus problem with the function f(x,y) = z, where z is the unstated dependent variable for the equation you gave for the third z-axis. Fortunately your two independent variables, x and y, aren't completely jumbled together. Conceptually it's the same approach as univariate calculus. The zeroes of the first partial derivatives will help identify maxima, minima, and saddle points, if there are any, where the slope of the tangent plane is zero. This involves finding partial derivatives of f(x,y) as part of the solution and it sets up a partial derivative vector called the gradient. A gradient is the same concept as a slope in three (or more) dimensions as a plane (in three dimensions) or a hyper-plane in four or more dimensions. When the set of all (x,y) are found that are zeroes of both the partial derivatives with respect to both x and y, it's called the set of zero vectors. Among them should be (if they exist) the global maximum, the global minimum, other local maxima, other local minima, and a phenomenon not found in univariate calculus, saddle points. Local maxima are other hilltops not as tall as the tallest peak in the mountain range. A saddle point looks like a saddle, and it's the low spot between two or more peaks that rises in their direction and falls off in the directions between the peaks.

See this video from the Kahn Academy and its follow-on videos to give you a conceptual idea of what you'll need to do . . .
https://www.khanacademy.org/math/multiva...and-minima

With regard to the HP Prime, look for the Gradient tools in its derivative calculus toolbox. Hope this helps get you started. The conceptual aspect of what you want to do is well over half the battle. You'll undoubtedly have to poke around a bit to find what you need to use and how to use it in the calculator. Looked for some things in the Prime documentation which isn't as helpful as it could be. Been decades since I did this stuff. All we had was pencils, engineering pads, a slide rule, and the CAS calculator was the CRC Handbook of Mathematical Tables (which has sections on derivatives and integrals). :-D

BTW, don't think of math involving three or more independent variables in hyperspaces as being that exotic. There are a multitude of practical applications for it in maximizing or minimizing aspects of physical processes and systems with multiple inputs or controls when responses are non-linear.

John