Calculus I MATH 131 Fall 2025

In section 3.3 we discovered that to find local (or relative) extreme values of a function (maximums and minimums), we must look for inputs at which the derivative was zero or undefined (these inputs are called critical numbers). This is a powerful tool that will be useful in applications.

However, we often are more interested in determining the global maximum or global minimum: the “optimal” value over all.

Do the first activity (3.5.2) from active calculus.

Must all functions have a global maximum or minimum? Well, a very simple function, \(f(x) = x\) does not. Any number you think is the maximum of the function will quickly be exceeded, just by increasing the input (and similarly for minimums, of course).

In practice though, we usually have some sort of constraint on the domain of the function. It might not make sense to allow inputs to be less than 0, for example. Or there might be a maximum allowable input. If we consider functions on closed intervals, then it turns out these always do attain their global maximum and minimum values (as long as the function is continuous on that interval).

Draw some pictures to illustrate this.

What does this mean in practical terms? The global maximum and minimum for a continuous function on a closed interval must occur at either a critical number or at one of the end points of the interval. So we really only have a handful of options. Which input gives us the largest/smallest output? Just try them all!

Find the global maximum and minimum of \(\dfrac{x^2}{x-2}\) on the interval \([3,7]\text{.}\)

How can we use global optimization to solve real world problems? Here is an example.

A piece of cardboard that is \(10\times 15\) (each measured in inches) is being made into a box without a top. To do so, squares are cut from each corner of the box and the remaining sides are folded up. If the box needs to be at least 1 inch deep and no more than 3 inches deep, what is the maximum possible volume of the box? what is the minimum volume? Justify your answers using calculus.