This problem is designed to help you achieve the following course objectives:

Find roots of equations using bracketing and open methods; explain the differences between the two methods and point out the advantage of each method.

Calculate minimum or maximum of a function.

Problem Description: Consider the following nonlinear function:

f(x)=x^4 + 2 x^2 - 5x + log(x+3)

(i) Plot f(x) in a figure to observe that it reaches a local minimum for some value of x in the range 1< x <1.

(ii) Compute the derivative, f'(x), plot it in the same figure, and observe that f' has a root in the range -1< x <1.

(iii) Observe that the root of f'(x) is exactly the value of x for which f(x) reaches its minimum.

(iv) Develop a MATLAB function to find the minimum of f(x) by finding the root of f'(x) using the Newton-Raphson method. Specifically, your function should take as input the error tolerance for the value of x* corresponding to the minimum of f(x). It should output both x* and the minimum function value f(x*). Read the function template for more details.

Given Code

function [xmin, fmin] = Minimize(es)

%% Input

% es: Stopping criterion for the approximate absolute *percent*

% relative error in xmin (scalar)

%% Output

% xmin: value of x at which function f(x) reaches minimum (-1 < xmin < 1)

% fmin: minimum of f(x), i.e. f(xmin).

% -------------------------

% TODO: Write the function body below

% Note: Recall that the Newton-Raphson method requires an initial guess.

% Choose an initial guess yourself.

f = @(x) _________________;

end