Plot multivariable function, find critical points

i need to plot a multivariable (x1,x2) function f_a in matlab, and find its critical points. I plotted it, but in order to find the critical points, i need to set the partial derivatives to zero then solve. when i try setting the partial derivatives to zero using diff(f_a,x1),it gives me an error.

[x1,x2] = meshgrid(-5:.2:5);                                
f_a = x1.^2 + x2.^2 +2.*x1.*x2;
figure(1)
surf(x1,x2,f_a)

fa1=diff(f_a,x1)

Answer 1

[Star Strider]

Use the gradient function to calculate the derivative.

Try this:

[x1,x2] = meshgrid(-5:.2:5);                                
f_a = x1.^2 + x2.^2 +2.*x1.*x2;
fa1 = gradient(f_a, 0.2, 0.2);                  % Derivative
zv = contour(x1,x2,fa1, [0; 0]);                % Critical Points

figure(1)
surf(x1,x2,f_a)
hold on
plot3(zv(1,2:end), zv(2,2:end), zeros(1,size(zv,2)-1), 'r', 'LineWidth',2)
hold off

[PJ]

thank you so much!

[Star Strider]

As always, my pleasure!

[PJ]

I tried it for another function and i'm not sure if it is giving me correct figures because there seems to be 3 red lines as contour lines, and I added another contour plot and found the critical points after, but the contour plot of figure 2 did not match the red lines of figure 1. I'm sure this is repetitive, but could you please take a look:

[x1,x2] = meshgrid(-5:.2:5);

f_c = x1.^4 + x2.^4 +1 -x1.^2+x2.^2;

figure(1)

surf(x1,x2,f_c)

[fc1,fc2] = gradient(f_c,0.2,0.2); %what do the 0.2s represent?

zv = contour(x1,x2,fc1, [0; 0]); %what do the 0s represent?

figure(1)

surf(x1,x2,f_c)

hold on

plot3(zv(1,2:end), zv(2,2:end), zeros(1,size(zv,2)-1), 'r', 'LineWidth',2 )

hold off

figure (2)

contour(x1,x2,f_c)

hold on

quiver(x1,x2,fc1,fc2)

hold off

syms x1 x2

fc1=diff(x1.^4 + x2.^4 +1 -x1.^2+x2.^2,x1);

fc2=diff(x1.^4 + x2.^4 +1 -x1.^2+x2.^2,x2);

[x1cr,x2cr]=solve(fc1,fc2);

critical_points=[x1cr,x2cr]

[Star Strider]

‘what do the 0.2s represent?’

They are the differences between the consecutive values of the meshgrid values in each direction.

‘what do the 0s represent?’

They specify that I want the contour function to trace out and return only the contours at 0.

I cannot follow what you are doing in the code you posted.