Finding Maximums and Minimums

We can try to find where a function $f$ is largest or smallest by evaluating $f$ at lots of values of $x$, a method which is not very efficient and may not find the exact place where $f$ achieves its extreme value. However, if we try hundreds or thousands of values for $\mathrm{x}$, then we can often find a value of $\mathrm{f}$ which is close to the maximum or minimum. In general, this type of exhaustive search is only practical if you have a computer do the work.

The graph of a function is a visual way of examining lots of values of $f$, and it is a good method, particularly if you have a computer to do the work for you. However, it is inefficient, and we still may not find the exact location of the maximum or minimum.

Calculus provides ways of drastically narrowing the number of points we need to examine to find the exact locations of maximums and minimums. Instead of examining $\mathrm{f}$ at thousands of values of $\mathrm{x}$, calculus can often guarantee that the maximum or minimum must occur at one of 3 or 4 values of $x$, a substantial improvement in efficiency.