Problem 4 - Calculus Early Transcendentals

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?

Step-by-Step Solution

Answer

Answer: One should look for the minimum value at the endpoints of the closed interval, as the single local maximum implies that the function will decrease on both sides of the maximum.

Step 1: Understanding the properties of a continuous function on a closed interval

Since the function is continuous on a closed interval, it means it takes on every value between its smallest and largest value within that interval. The interval is closed, which means it includes its endpoints.

Step 2: Analyzing the local maximum

We have information that the function has only one local maximum. This means that the function increases up to the local maximum and then decreases afterwards. On the other side of the maximum, if we move further away from it, the function would decrease.

Step 3: Identifying potential minimum points

Keeping in mind the behavior of the function around the local maximum, we can conclude that the minimum value of the function must be at either of the two endpoints of the interval (i.e., the start or the end of the interval).

Step 4: Comparing the function values at the endpoints

In order to find where the minimum value lies, we need to calculate the function values at the two endpoints of the interval. Whichever endpoint has a lower function value will be considered the minimum of the function on the given closed interval.

Step 5: Conclusion

To find the solution to the problem, one should look for the minimum value of the continuous objective function at the endpoints of the closed interval, as the single local maximum implies that the function will decrease on both sides of the maximum.