Global Maxima & Minima
Learn global maxima and minima for NSW Year 12 Mathematics Advanced (NESA outcome MAV-12-06). The overall greatest or least value of a function on a domain is not always at a turning point; it may sit at an endpoint or at a corner where the derivative does not exist.
You will learn to find stationary points inside the domain, evaluate the function there and at each endpoint, compare the values to name the global maximum and minimum, and tell them apart from local ones — the checking step every optimisation answer needs.
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.