Optimization

Steps for Solving Optimization Problems

1. Write the function that is to be maximized or minimized.

2. Write an equation relating the different variables of the problem, in the case that there is more than one variable.

3.Work out an equation for one of the variables and replace it in the function so that there is only one variable remaining.

4. Differentiate the function and equate to zero to find the local extrema.

5. Calculate the 2nd derivative and verify the result.

Example

Determine the length of the sides of an isosceles triangle measuring 12 metres in perimetre that maximize its area.

Isosceles Triangle

The function that needs to be maximized is the area of the triangle:

Optimization

Relate the variables:

2x + 2y = 12

x = 6 − y

Substitute in the function:

Optimization Operations

Differentiate, equate to zero and calculate the roots.

Optimization Operations

Optimization Operations

Determine the 2nd derivative and replace by 2. Discard the solution y = 0 because there is no triangle which has a side of zero.

Optimization Operations

Optimization Operations

Optimization Operations

Therefore it remains proven that at y = 2 there is a maximum.

The base (2y) measures 4m and the side obliques (x) also are 4 m, so the triangle of maximum area would be an equilateral triangle.