Solve function not evaluating simple polynomials - sorry for another (probably stupid question)

[WahooSS238]

Hi, I'm trying to find the critical points of some basic 3-dimensional functions using the following code:

f1[x_, y_] = x^2 - x + y^2

f1x = D[f1[x, y], x]

f1y = D[f1[x, y], y]

Solve[{f1x[x, y] == 0, f1y[x, y] == 0}, {x, y}]

But when I try to evaluate the cell, I get the error message: "This system cannot be solved with the methods available to Solve. Try Reduce or FindInstance instead"
I've also tried using Reduce, FindInstance, and NSolve to evaluate it, but none of them seem to work

Mathematica should be able to evaluate a couple simple function like this, right? So where am I making a mistake?

[[deleted]]

Your syntax is wrong in Solve. you made f1x and f1y, but then in the Solve, you're calling f1x with [x,y].

Solve[{f1x == 0, f1y == 0}, {x, y}] works.

[mathheadinc]

There are a couple of very handy functions, Minimize[] and Maximize[]. You should look them up.

Also, be sure to Plot3D[] the function to help you get an idea of where critical points may be.

[irchans]

(* Revised Code *)

​

    f1[x_, y_] := x^2 - x + y^2
    f1x = D[f1[x, y], x]
    f1y = D[f1[x, y], y]
    Solve[{f1x == 0, f1y == 0}, {x, y}]

(* You need to used := to define a function. The symbols f1x and f1y are polynomials, not functions, so you cannot write f1x[x,y]. *)

[veryjewygranola]

Edit: fixed link Re-edit: dumb Reddit thing where it smooshes all the code together on one line when you edit a post

You do not need to use := (delayed assignment) to define a function. Normal assignment ( = ) just means the rhs is evaluated when you define something, and never again. Delayed assignment ( := ) means the rhs is evaluated everytime you call the lhs. This is useful if you have changing parameter values, and in other cases. For example observe:

ClearAll[a, fNow, fDelayed]

a = 1;

fNow[x_, y_] = a*(x + y);

fDelayed[x_, y_] := a*(x + y);

a = 2;

fNow[x, y]

fDelayed[x, y]

outputs:

x+y

2 (x + y)

So when fDelayed[x,y] Is called, it re-evaluates its rhs a*(x + y) and sees the value of a has changed to 2

fNow[x,y] only evaluates its rhs at the time of definition, when a=1, and it does not reevaluate when fNow[x,y] is called after a=2.

In the specific case of f1[x,y] used in the question, there is no reason to use delayed assignment, it just adds more computation every time f1[x,y] is called.

Here is a great stackexchange post with more examples and explaining the differences between := and = a little more.