Final answer:
To find the minimum of the function f(x)=(x²/12)−2sin(x), write its derivative and use numerical methods in an M-file to find the root of the derivative, which indicates the critical point of the function. Starting from the initial guesses, iterate until you find a minimum within a certain tolerance.
Step-by-step explanation:
To find the minimum of the function f(x)=(x²/12)−2sin(x) using an M-file, we can use numerical methods like Newton's method or the secant method for finding a root of the derivative of the function, as the root of the derivative indicates where the function might have a minimum (critical point).
First, we need to write the derivative of the function, which is f'(x) = (x/6) - 2cos(x). Then, we would write an M-file in MATLAB that implements the chosen numerical method. The M-file will start from initial guesses x1=0 and x3=4 to calculate subsequent guesses and iterate until it converges to a solution that is within a predefined tolerance.
An example of such an algorithm could involve initializing two values, x1 and x3, computing new points based on the derivative and the previous points until we find a point where the derivative is close to zero. That point is likely to be where the function has a minimum unless it is an inflection point or a maximum, which can be determined by checking the second derivative.