finding local maximum/minimum for a function

how to find the local extreme values of the function xy-x^2-y^2-2x-2y+4

[Adam Danz]

Some starting points:

• https://www.mathworks.com/help/symbolic/maxima-minima-and-inflection-points.html
• https://www.mathworks.com/help/matlab/ref/islocalmax.html
• https://www.mathworks.com/help/signal/ref/findpeaks.html

Answer 1

[sanidhyak]

Hi Adonia,

I understand that you want to find the local maximum and local minimum of this function: xy-x^2-y^2-2x-2y+4

You can follow the following steps:

• Define the function and compute first-order partial derivatives using “diff”.
• Solve for critical points using “solve” and “compute” second-order partial derivatives.
• Calculate the Hessian determinant and use it to classify the critical points (min, max, or saddle) based on its value and the second derivative.

Use the "plot" function in MATLAB to visualize the function's surface and identify the critical points on it. This will help you to verify their nature based on the graphical representation.

% Plot the function 
fsurf(f, [-5 5 -5 5]) % Plot the surface 
hold on 

% Plot critical points in red 
plot3(double(crit.x), double(crit.y), double(subs(f, [x, y], [crit.x, crit.y])), 'ro', 'MarkerSize', 8, 'LineWidth', 2) 
xlabel('x'), ylabel('y'), zlabel('f(x, y)') 
title('Function Plot with Critical Points') 
grid on 
hold off

Kindly refer to the following MATLAB documentation for detailed information on all the related functions

• https://www.mathworks.com/help/symbolic/maxima-minima-and-inflection-points.html