# 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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#### Solution Preview

To use Excel for this question (part a), you need to set up two columns of data. The first column, label it X, is the value of x where you evaluate the expression. The second column, label it f(x) is where you find the functions value at the point X. Please see the attached excel spreadsheet for ...

#### Solution Summary

The solution clearly explains how to solve for the minimum and maximum plus the derivative of f(x). The commands in Excel at included, as well as a graph is easier understanding.