Every portfolio optimisation problem has the same mathematical skeleton: minimise or maximise an objective function over a feasible set. Before asking how to find the optimum, we must ask whether an optimum exists. The answer is the Extreme Value Theorem: on a closed, bounded domain, every continuous function attains both its maximum and minimum values. This guarantee is non-trivial — remove either condition (continuity or the closed interval) and the optimum can vanish entirely. Once existence is secured, the derivative narrows the search: any interior optimum must occur at a point where is zero or undefined. The closed-interval method collects all these candidates, evaluates at each, and reads off the global extremum by comparison. This is the foundational procedure behind profit maximisation, minimum-variance portfolio construction, and the Markowitz efficient frontier.
Learning objectives. By the end of this note you should be able to:
- distinguish absolute (global) extrema from local (relative) extrema;
- state the Extreme Value Theorem precisely and identify when each hypothesis is necessary;
- apply Fermat's Theorem to narrow the search to critical numbers, and avoid the two classic counterexample errors;
- execute the three-step closed-interval method to find absolute extrema on ;
- interpret the critical-point condition as the profit-maximisation condition marginal revenue = marginal cost.
In this lesson
- Absolute and local extrema
- The Extreme Value Theorem
- Fermat's Theorem and critical numbers
- The Closed-Interval Method
Before you start
- Continuity, the Intermediate Value Theorem, and the Extreme Value Theorem
- The Derivative as a Limit: Rates of Change and the Derivative Function
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