How do you find the maximum and minimum of a function on an interval?

Before we calculate anything, we rely on a mathematical guarantee called the Extreme Value Theorem. This theorem states that if a function is continuous on a closed interval $[a, b]$, it must have both an absolute maximum and an absolute minimum on that interval. A closed interval means the endpoints are included. As long as there are no breaks or holes in the graph, we are guaranteed to find a highest and lowest point.

Critical points are the "peaks and valleys" of our function. To find them, we take the first derivative of the function, $f'(x)$, and find where it is either equal to zero or undefined. When the derivative is zero, the tangent line to the graph is perfectly flat, which usually indicates a local peak or valley. We only care about the critical points that actually fall inside our specific interval $[a, b]$.

The step-by-step process for finding these extremes is called the Closed Interval Method. First, find the derivative of the function. Second, set the derivative to zero or find where it is undefined to get your critical points. Third, plug those critical points into the original function $f(x)$ to get their $y$-values. Finally, plug the endpoints $a$ and $b$ into the original function. Compare all the resulting $y$-values: the largest is your maximum, and the smallest is your minimum.

A common mistake is plugging the critical points back into the derivative instead of the original function. Remember, the derivative just tells us the slope, but we are looking for the actual height ($y$-value) of the function. Another frequent error is forgetting to check the endpoints. Often, the highest or lowest point of a specific interval isn't a peak or valley at all, but rather the very edge of the graph.