Global Maxima & Minima
Global maxima and minima find the largest and smallest values of a function on an interval by comparing stationary points with the endpoints.
Part of the NSW Year 12 Mathematics Advanced course, in the Calculus area of study (Applications of calculus focus area) of the 2024 syllabus. Work through practice questions with fully worked solutions and video lessons, or scroll down for the theory summary and worked examples.
Theory
On a closed interval, the greatest and least values occur at a stationary point or an endpoint. This Year 12 Mathematics Advanced topic (MAV-12-06) finds global extremes.
To find the global max/min of \(f\) on \([a,b]\): find the stationary points inside (\(f'=0\)), evaluate \(f\) there and at the endpoints \(a,b\), then compare the \(y\)-values.
Method
- Find stationary points inside the interval.
- Evaluate \(f\) there and at both endpoints.
- Compare the values: largest is the max, smallest the min.
| \(2x-4\) | \(=\) | \(0\Rightarrow x=2\) |
| \(f(2)\) | \(=\) | \(-1\) |
| \(f(0)\) | \(=\) | \(3\) |
| \(f(3)\) | \(=\) | \(0\) |
| \(\text{max}\) | \(=\) | \(3\ (x=0)\) |
| \(\text{min}\) | \(=\) | \(-1\ (x=2)\) |
| \(f(1)\) | \(=\) | \(1\) |
| \(f(4)\) | \(=\) | \(16\ (\text{max})\) |
Common pitfalls
Frequently asked questions
How do you find the global maximum on a closed interval?
Evaluate the function at every stationary point inside the interval and at both endpoints, then pick the largest value.
Why check the endpoints?
On a closed interval the greatest or least value is often at an end, not at a turning point.
What is the difference between local and global extremes?
A local extreme is highest/lowest nearby; a global extreme is highest/lowest over the whole interval.
What if there is no stationary point in the interval?
Then the global max and min are both at the endpoints.