Hi @xin,

You've defined a symbolic function involving several variables and taken its derivative with respect to rho. The goal is to find the critical points where the derivative equals zero and to determine whether these points correspond to maxima or minima using the second derivative test. In addition to your comments, I did add code to visualize its derivative to better understand the behavior over a specified range. Below is the complete MATLAB code that fulfills your requirements:

% Define symbolic variables
syms Vr theta_r Vse rho gamma Ish positive real
syms X_r positive real
syms n_t positive real
syms Vs positive real

% Define the expression
expr = sqrt(((3*Vs)/2 + Vse*n_t*cos(rho) - Vr*n_t*cos(theta_r))^2/
(X_r^2*n_t^2) + ...
((sqrt(3)*Vs)/2 + Vse*n_t*sin(rho) - Vr*n_t*sin(theta_r))^2/
(X_r^2*n_t^2));

% Compute the derivative with respect to rho
diff_expr = simplify(diff(expr, rho));

% Solve for critical points where derivative equals zero with conditions
critical_points = solve(diff_expr == 0, rho, 'ReturnConditions', true);

% Display critical points
disp('Critical points:');
disp(critical_points);

% Compute the second derivative
second_diff_expr = simplify(diff(diff_expr, rho));

% Evaluate the second derivative at critical points
second_derivative_values = subs(second_diff_expr, rho, critical_points.rho);

% Analyze the nature of critical points
disp('Second derivative values at critical points:');
disp(second_derivative_values);

% Create a range for rho for plotting
rho_values = linspace(0, 2*pi, 100); % Adjust range as needed

% Substitute numeric values for symbolic variables (example values)
Vr_val = 1; % Assign a numeric value
theta_r_val = 0.5; % Assign a numeric value
Vse_val = 1; % Assign a numeric value
X_r_val = 1; % Assign a numeric value
n_t_val = 1; % Assign a numeric value
Vs_val = 1; % Assign a numeric value

% Evaluate the expression and derivative numerically
% Using the correct syntax to substitute multiple variables
expr_values = subs(expr, {Vr, theta_r, Vse, X_r, n_t, Vs}, {Vr_val, theta_r_val,
Vse_val, X_r_val, n_t_val, Vs_val});
diff_values = subs(diff_expr, {Vr, theta_r, Vse, X_r, n_t, Vs}, {Vr_val,
theta_r_val, Vse_val, X_r_val, n_t_val, Vs_val});

% Convert symbolic results to double for plotting
expr_values_numeric = double(subs(expr_values, rho, rho_values));
diff_values_numeric = double(subs(diff_values, rho, rho_values));

% Plot the function
figure;
plot(rho_values, expr_values_numeric, 'LineWidth', 2);
xlabel('rho');
ylabel('Function Value');
title('Plot of the Function over rho');
grid on;

% Optionally, plot the derivative
figure;
plot(rho_values, diff_values_numeric, 'LineWidth', 2);
xlabel('rho');
ylabel('Derivative Value');
title('Plot of the Derivative over rho');
grid on;

Please see attached.

**Explanation of the Code**

Symbolic Variable Definition: The required symbolic variables are defined with appropriate constraints (e.g., positive real).

Expression Definition:The expression you provided is defined symbolically.

First Derivative:The derivative of the expression with respect to \( \rho \) is computed and simplified.

Critical Points:The critical points where the derivative equals zero are calculated using the `solve` function.

Second Derivative: The second derivative is computed to analyze the concavity at the critical points, which helps in determining whether they correspond to maxima or minima.

Evaluation of Second Derivative:The second derivative is evaluated at each critical point to ascertain the nature of those points.

Visualization: The function and its derivative are plotted over a specified range of rho values for visual analysis.

Now, if the second derivative at a critical point is positive, that point corresponds to a local minimum and if it is negative,then that indicates a local maximum. If it equals zero, further tests may be needed. Also, adjust the range of rho values based on the specific context of your problem to make sure you capture the relevant behavior of the function. When applying this code, you'll need to assign numerical values to the symbolic variables (e.g., Vr, Vs, etc.) to generate specific numerical results.

Please let me know if this helped resolve your problem.