Concavity

CHECK IT

clc;
clear all;

% Define the symbolic variable
syms x real;

% Input the function from the user
f = input("Enter the function f(x): ");

% Compute the first derivative
fx = diff(f, x);
critical_points = solve(fx);

% Calculate min and max of critical points for plotting
cmin = min(double(critical_points));
cmax = max(double(critical_points));

% Plot the original function
figure(1);
ezplot(f, [cmin-2, cmax+2]);
hold on;

% Compute the second derivative
fxx = diff(fx, x);

% Analyze critical points for maxima, minima, and inflection points
for i = 1:length(critical_points)
    second_derivative_value = subs(fxx, x, critical_points(i));
    function_value = subs(f, x, critical_points(i));
    
    if double(second_derivative_value) == 0
        fprintf("The point x = %.2f is an inflection point\n", double(critical_points(i)));
    elseif double(second_derivative_value) < 0
        fprintf("The maximum point x = %.2f\n", double(critical_points(i)));
        fprintf("The value of the function is %.2f\n", double(function_value));
    else
        fprintf("The minimum point x = %.2f\n", double(critical_points(i)));
        fprintf("The value of the function is %.2f\n", double(function_value));
    end
    
    plot(double(critical_points(i)), double(function_value), "r*", "MarkerSize", 15);
end

% Identify and plot inflection points
degree = polynomialDegree(fxx);
if degree == 0
    fprintf("The given polynomial is second degree or less\n");
else
    inflection_points = solve(fxx); % Find inflection points
    for i = 1:length(inflection_points)
        inflection_value = subs(f, x, inflection_points(i));
        plot(double(inflection_points(i)), double(inflection_value), "g*", "MarkerSize", 15);
    end
end

% Plot the first derivative
figure(2);
ezplot(fx, [cmin-2, cmax+2]);
title("Plot of the First Derivative of f and Critical Points");
hold on;

for i = 1:length(critical_points)
    critical_value = subs(fx, x, critical_points(i));
    plot(double(critical_points(i)), double(critical_value), "r*", "MarkerSize", 15);
end

% Plot the second derivative
figure(3);
ezplot(fxx, [cmin-2, cmax+2]);
hold on;

if degree == 0
    fprintf("The given polynomial is second degree or less; second derivative plot is not possible.\n");
else
    for i = 1:length(inflection_points)
        second_derivative_value = subs(fxx, x, inflection_points(i));
        plot(double(inflection_points(i)), double(second_derivative_value), "r*", "MarkerSize", 15);
    end
    title("Plot of the Second Derivative of f and Inflection Points");
end