among the locations at which the function f(x)=cos(121/64x2)−sin(11/4x) has a local minimum, the smallest in absolute value is: in matlab

To find the local minimum of a function in MATLAB, we can use the fminsearch function. Since we are working with a 1D function, we just need to provide an initial guess for the minimum location.

Here's an example code to find the smallest local minimum of the given function:

main.m
% Define the function
f = @(x) cos(121/64*x^2) - sin(11/4*x);

% Initial guess for the minimum location
x0 = 0;

% Use fminsearch to find the minimum location
xmin = fminsearch(f, x0);

% Check if the minimum is indeed a local minimum
islocalmin = abs(fminbnd(f, xmin-1, xmin+1) - f(xmin)) < 1e-6;

% Display the result
if islocalmin
    fprintf('The smallest local minimum is at x = %f\n', xmin)
else
    fprintf('No local minimum found\n')
end
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Note that we use fminbnd to check if the found minimum is indeed a local minimum within a small interval around it. The tolerance of 1e-6 is chosen arbirarily, you might want to adjust it to your needs.