# Minimum and maximum of a function that is not constrained

To find the minimum or maximum of a function that is not constrained, that is, there are no restrictions on range of the function, you take the derivative of the function, set it equal to zero, and solve for the variable. The intuition is that the derivative is the instantaneous slope of the function. A minimum or maximum occurs when the instantaneous slope equals 0.

a) Using Excel, find the minimum, x*, of f(x) = x2 + 2. What is f(x*)? Calculate f(x) for x = -2 to 2 at 0.1 intervals. Plot f(x).

b) Calculate the derivative of f(x), f'(x). Set it equal to zero and solve for x.

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Once this is accomplished, you can use the chart wizard to plot the data. Again, see the excel spreadsheet for help.

Now that we have the data and the chart, it is easy to see from either source that the minimum of the function in this range is at x*=0. At this point, ...