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IB DP Maths AA SL Study Notes

5.3.2 Global Extrema

What are Global Extrema?

Global extrema are the absolute highest (maximum) or lowest (minimum) values a function can achieve over its entire domain. This stands in contrast to local extrema, which represent high or low points within a specific interval or vicinity of a point.

  • Global Maximum: This represents the zenith, the highest value a function can achieve across its entire domain.
  • Global Minimum: On the flip side, this is the nadir, the lowest value a function can reach.

For any function f(x), if there's a value f(c) that stands out as the global maximum or minimum over its domain D, then this value is the global extremum. This peak or trough occurs at the point x = c.

Techniques to Pinpoint Global Extrema

Identifying global extrema requires a methodical approach:

1. Spot the Critical Points: Begin by identifying the critical points of the function. These are where the derivative is zero or doesn't exist. They're potential candidates for extrema.

2. Evaluate the Endpoints: If the function's domain is a closed interval (with defined start and end points), evaluate the function at these boundaries. Sometimes, the global extrema might be found here.

3. Compare Values: With the function's values at the critical points and endpoints in hand, compare them. The highest is the global maximum, and the lowest is the global minimum.

The Real-world Importance of Global Extrema

Global extrema have profound implications beyond academic interest:

  • Economics: Businesses often turn to calculus to determine the best production level to maximise profit or minimise costs. Here, the global maximum could represent the highest possible profit, while the global minimum might indicate the lowest costs.
  • Engineering: Engineers use calculus to determine the best dimensions for a structure, ensuring maximum strength or efficiency.
  • Environmental Science: In studies of population dynamics or resource consumption, the global maximum might represent an environment's carrying capacity.

Diving In with Examples

Example 1: Determine the global extrema of the function f(x) = x2 - 4x + 4 over the interval [0, 3].

Solution: Start by finding the derivative: f'(x) = 2x - 4. Setting this to zero gives x = 2 as the critical point. Now, evaluate the function at this point and the interval's endpoints:

  • f(0) = 4
  • f(2) = 0
  • f(3) = 1

From this, the global minimum is 0 at x = 2, and the global maximum is 4 at x = 0.

Example 2: A company's profit function, denoted as P(x), from producing x units of a product is P(x) = -x2 + 40x - 200. What production level maximises profit?

Solution: Differentiate the profit function to get P'(x) = -2x + 40. Setting this to zero, x = 20 emerges as the critical point. Analysing the function further, this point is a global maximum. Thus, for maximum profit, the company should produce 20 units.

FAQ

Yes, there are functions that don't have global extrema. For instance, functions that are always increasing or always decreasing over their entire domain don't have both a global maximum and a global minimum. The function f(x) = x is an example, as it's always increasing and doesn't have a highest or lowest point over its domain of all real numbers. Similarly, functions that oscillate infinitely, like sine and cosine, don't have global extrema over an infinite domain.

The second derivative test is primarily used to determine the concavity of a function and to identify local extrema. If the second derivative at a point is positive, the function is concave upwards at that point, indicating a local minimum. If it's negative, the function is concave downwards, indicating a local maximum. However, the second derivative test doesn't directly provide information about global extrema. To determine global extrema, especially on a closed interval, one must evaluate the function at critical points and endpoints and compare the values.

When a function is defined over a closed interval, the endpoints play a crucial role in determining global extrema. After finding the critical points, it's essential to evaluate the function at these endpoints. The reason is that the global maximum or minimum might occur at these boundaries. By comparing the function values at the critical points and the endpoints, one can determine the global extrema. If the domain is not a closed interval, then the function might not have global extrema.

A function can have at most one global maximum and one global minimum. However, it's possible for a function to have neither a global maximum nor a global minimum. It's also possible for a function to have just one of them. For instance, a function that is always increasing will have a global minimum but no global maximum. Conversely, a function that is always decreasing will have a global maximum but no global minimum.

Global extrema refer to the absolute highest or lowest values a function can achieve over its entire domain. In other words, a global maximum is the highest point of the function across all x-values, and similarly, a global minimum is the lowest point across all x-values. On the other hand, local extrema are the high or low points within a specific interval or vicinity of a point. A local maximum might not be the highest point of the entire function but is the highest within its immediate surroundings. Similarly, a local minimum is the lowest point in its immediate vicinity but not necessarily the lowest point of the entire function.

Practice Questions

A company produces and sells a product. The profit function, P(x), from producing and selling x units of the product is given by P(x) = -2x^2 + 50x - 120. Determine the production level that maximises the profit.

To find the production level that maximises profit, we need to determine the global maximum of the profit function. Start by finding the derivative of P(x) with respect to x. Differentiating, we get P'(x) = -4x + 50. Setting this to zero, we get x = 12.5. This means that the company should produce and sell 12.5 units to maximise profit. However, since the company cannot produce half a unit, the optimal production level is either 12 or 13 units. By evaluating the profit function at these points, we can determine that producing 12 units yields the maximum profit.

A rectangular garden is to be fenced off with one of its sides against a wall. If only 60 metres of fencing is available, determine the dimensions of the garden that will maximise its area.

Let the length of the garden perpendicular to the wall be x metres. Since one side is against the wall, the length of the garden parallel to the wall will be (60 - 2x) metres. The area, A, of the garden is given by A(x) = x(60 - 2x) = 60x - 2x2. To maximise the area, differentiate A with respect to x. Differentiating, we get A'(x) = 60 - 4x. Setting this to zero, we find x = 15. Thus, the garden should be 15 metres in length perpendicular to the wall and 30 metres in length parallel to the wall to maximise its area.

Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
LinkedIn
Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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