Implement a Matlab function which completes the following task: Give the minimax approximation of a given function f on (a a closed interval; (b)a set of finitely many points.

Here is an implementation of a Matlab function that calculates the minimax approximation of a given function f on a closed interval [a,b] using a set of finitely many points:

function [p, err] = minimax(f, a, b, n)
% MINIMAX Approximation of f on [a,b] with n points using minimax method
% p is the polynomial and err is the maximum error

% Create Chebyshev nodes
x = cos((0:n-1)*pi/(n-1))';
x = 0.5*(a+b) + 0.5*(b-a)*x;

% Calculate Chebyshev coefficients
c = chebcoeffs(f, x);

% Construct minimax polynomial
p = chebval(x, c);

% Calculate maximum error
err = max(abs(f(x) - p));
end

function c = chebcoeffs(f, x)
% CHEBCOEFFS Calculate Chebyshev coefficients of f at nodes x

% Chebyshev polynomials
n = length(x);
T = zeros(n,n);
T(:,1) = ones(n,1);
T(:,2) = x;

for k = 3:n
    T(:,k) = 2*x.*T(:,k-1) - T(:,k-2);
end

% Inner product matrix
A = T'*T;

% Right-hand side vector
b = T'*f(x);

% Chebyshev coefficients
c = A\b;
end

To use this function, simply provide the function `f`, the interval endpoints `a` and `b`, and the number of points `n` to use in the approximation. The function returns the minimax polynomial `p` and the maximum error `err`.

Note that this implementation uses Chebyshev nodes and Chebyshev polynomials to construct the minimax polynomial. Chebyshev nodes are known to give the best possible approximation in the minimax sense, and Chebyshev polynomials are used as the basis functions because they are orthogonal with respect to the L2 inner product.