Question

Suppose you wish to minimize a continuous objective function on a closed interval, but you find that it has only a single local maximum. Where should you look for the solution to the problem?

Short answer

Answer: In a continuous function on a closed interval with only one local maximum, we should look for the solution to the problem of finding the point where the function is minimized at the endpoints of the closed interval.

Step-by-step solution

Understand the properties of a continuous function on a closed interval

According to the Extreme Value Theorem, if a function is continuous on a closed interval, it must attain both its minimum and maximum values on that interval. As the given function has only one local maximum, we can conclude there must be at least two points on the interval that are minimum, which are at the endpoints.

Find the local maximum

As the function has only one local maximum, differentiate the function regarding the variable to obtain the first derivative. Then, find the critical point where the first derivative is equal to 0 or does not exist. This will lead to the local maximum.

Compare the function's values at the local maximum and endpoints

Once the local maximum and the endpoints are established, compute the function's values at those points. The minimum value of the function will be the lowest among the values at the local maximum and the endpoints.

Determine the solution to the problem

The solution to the problem (where the function is minimized) will be at the point with the smallest value among the local maximum and the endpoints. As the function has only one local maximum, the solution must be at one of the endpoints of the closed interval.

Key concepts

Continuous Objective Function

When we talk about optimizing a continuous objective function in calculus, we refer to a function that smoothly varies without any abrupt changes or breaks within its domain.

For instance, when you're tasked with minimizing costs, maximizing efficiency, or finding an optimal design within certain parameters, you're likely dealing with a continuous objective function. These functions are predictable and follow a smooth curve when graphed. An essential characteristic of such functions is that they can be graphed without lifting your pencil off the paper, symbolizing their uninterrupted nature.

Importance in Optimization

In optimization, especially within a closed interval, understanding that a function is continuous is key. This characteristic ensures that the search for optimal values (minimum or maximum) does not need to consider sudden jumps or undefined regions within the interval. It allows for comprehensive methods like calculus-based optimization to find the solution.

When approaching such a problem, the continuity plays a pivotal role because certain theorems and principles only apply to continuous functions, leading to reliable and predictable results.

Closed Interval Optimization

Optimization over a closed interval means that you are looking to find the minimum or maximum value of a function within a specific, finite range.

This range is inclusive of its endpoints, which is why it's referred to as 'closed'. It's like searching for the best solution within a well-defined box of possibilities, without stepping outside the boundaries.

Relevance to Finding Extremes

The closed nature of the interval ensures that we always include the boundary values in our considerations. This is crucial because the extreme values of a function—where it reaches its highest or lowest points—can sometimes occur right at the edges of the domain we're looking at.

Consequently, when you are asked to minimize a function on such an interval, it's essential to check the function’s value at these boundary points, as they could potentially hold the answer to the problem.

Extreme Value Theorem

The Extreme Value Theorem is a fundamental result in calculus that assures us of the existence of both a maximum and minimum value for a continuous function over a closed interval.

Simply put, if you have a function that doesn't break, pause, or leap within a certain range, the theorem guarantees that there are specific points where the function reaches its highest and lowest values.

Applying the Theorem

When addressing an optimization problem where it's given that the function has only one local maximum and seemingly no minimum, the theorem instructs us to also examine the endpoints of the interval.

In practice, after identifying any critical points—where the derivative is zero or undefined—compare the value of the function at these points to its value at the endpoints of the range. Among these, the smallest value will be the absolute minimum on the closed interval, thanks to the assurance provided by the Extreme Value Theorem.