How to find extrema using Matlab symbolic operations

You need to download the file ezcontourc.m.

Example: Find all relative maxima and minima of f(x,y) = x³ - 3x + 3xy²

Define the symbolic variables and f

syms x y
f = x^3 - 3*x + 3*x*y^2

Find the partial derivatives

fx = diff(f,x)
fy = diff(f,y)

Find critical points (xc,yc) by solving fx=0 and fy=0 for x and y

[xc,yc] = solve(fx,fy,x,y); [xc,yc]

Matlab finds 4 solutions. The first solution is xc(1),yc(1),..., the 4th solution is xc(4),yc(4).

Find the second partials and the discriminant D

fxx = diff(fx,x); fxy = diff(fx,y); fyy = diff(fy,y)
D = fxx*fyy - fxy^2

Evaluate D at the first critical point by substituting for x and y the values xc(1) and yc(1)

subs(D,{x,y},{xc(1),yc(1)})
subs(fxx,{x,y},{xc(1),yc(1)})

Note that we have D > 0 and fxx > 0, hence this is a relative minimum.

We can similarly check the other three critical points and find that two of them are saddle points and one is relative maximum.

Plot the function as a surface and as contours (with 51 contours)

figure(1); ezsurf(f,[-1.5,1.5,-1.5,1.5]);
figure(2); ezcontourc(f,[-1.5,1.5,-1.5,1.5],51); axis equal; axis tight