What Is an Optimization Problem and How Do You Solve One?

An optimization problem asks you to find the greatest or least possible value of a quantity, usually subject to a constraint. In IB Mathematics: Analysis and Approaches SL 5.8, you solve it by creating a one-variable function, differentiating it, and proving where its maximum or minimum occurs.

The Optimization Method

The quantity being maximized or minimized is the objective function. A constraint connects the variables and allows the objective function to be written using only one variable. Its feasible domain contains values that satisfy the context, such as positive lengths or a limited amount of material.

For example, a rectangle has perimeter $20\text{ m}$. If its sides are $x$ and $y$, then

$$2x+2y=20 \quad \Rightarrow \quad y=10-x.$$

Its area is therefore

$$A(x)=xy=x(10-x)=10x-x^2.$$

Differentiate and set the derivative equal to zero:

$$A'(x)=10-2x=0 \quad \Rightarrow \quad x=5.$$

Since $A''(x)=-2<0$, this stationary point is a maximum. Thus $y=5$, and the maximum area is $25\text{ m}^2$. Here, the practical domain is . The derivative changes from positive to negative at , which independently confirms the maximum and connects the algebra to the graph.

Exam Technique

A common misconception is that $f'(x)=0$ automatically proves an optimum. It does not: classify the stationary point and check endpoints when the domain is closed. On Paper 1, show exact analytic working; on Paper 2, write the function and derivative equation before using your GDC. State the optimized quantity with units, not only the variable value.