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The solution to finding the local and global extrema of two given functions using matlab. The matlab code to evaluate and visualize the functions and their critical points, as well as the output of the code for the critical points and extrema. The first function is x^3-12x-5 in the interval [-4,4], and the second function is x^2*exp(sin(x))-x/x^3+1 in the interval [0,5]
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clear all clc syms x f = input ('Enter the function f(x):'); I = input ('Enter the interval: '); a=I (1); b=I (2); df = diff (f, x); ddf = diff (df, x); f = inline(vectorize(f)); df = inline(vectorize(df)); ddf = inline(vectorize(ddf)); range = linspace (a, b,100); plot (range, f(range),'-b','LineWidth',2); legstr = {'Function Plot'}; % Legend String hold on; guesses = linspace (a, b,5); root = zeros(size(guesses)); for i=1: numel(guesses) root(i) = fzero (df, guesses(i)); end root = root (a <= root & root <=b); root = unique(round(root,4)); plot (root, f(root),'ro','MarkerSize',10); legstr = [legstr, {'Critical Points'}]; disp (['Critical Points of f(x) are: ', num2str(root)]) maxp = root(ddf(root) < 0); if(numel(maxp) ~= 0) disp (['Local maximum of f(x) occurs at: ', num2str(maxp)]) end minp = root(ddf(root) > 0); if(numel(minp) ~= 0) disp (['Local minimum of f(x) occurs at: ', num2str(minp)]) end fval = f(root); if(numel(maxp) ~= 0) gmax = root (fval == max(fval)); disp (['Global maximum of f(x) occurs at: ', num2str(gmax),' and its value is:', num2str(max(fval))]) plot (gmax, f(gmax),'m+','MarkerSize',10); legstr = [legstr, {'Global Maximum'}]; end if(numel(minp) ~= 0) gmin = root (fval == min(fval)); disp (['Global minimum of f(x) occurs at: ', num2str(gmin),' and its value is: ', num2str(min(fval))]) plot (gmin, f(gmin),'m*','MarkerSize',10); legstr = [legstr, {'Global Minimum'}]; end legend (legstr, 'Location', 'Best')
2. Find the global extrema of the function f(x)=x^2 esinx- x x 3
clear all clc syms x f = input ('Enter the function f(x):'); I = input ('Enter the interval: '); a=I (1); b=I (2); df = diff (f, x); ddf = diff (df, x); f = inline(vectorize(f)); df = inline(vectorize(df)); ddf = inline(vectorize(ddf)); range = linspace (a, b,100); plot (range, f(range),'-b','LineWidth',2); legstr = {'Function Plot'}; % Legend String hold on; guesses = linspace (a, b,5); root = zeros(size(guesses)); for i=1: numel(guesses) root(i) = fzero (df, guesses(i)); end root = root (a <= root & root <=b); root = unique(round(root,4));