Question

True or False? In Exercises \(53-56,\) determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The maximum of a function that is continuous on a closed interval can occur at two different values in the interval.

Short answer

The statement is True. A function continuous on a closed interval can have a maximum occurring at two different values.

Step-by-step solution

Understand the problem

The problem statement is asking if the maximum of a function that is continuous on a closed interval can occur at two different values within the interval. This is about determining the maximum value of a function, which in many cases only occurs at one point. But could there be exceptions?

Consider the extremes

Consider the extremal values of a function. Should we consider a function like \(f(x) = x^{2}\), for example, the maximum would be at just a single value. However, for a function like \(f(x) = -x^{2}\), on the closed interval [-1,1], the maximum would be at two different points, i.e., x= -1 and x= 1. Therefore an exception may exist.

Make a Conclusion

Based on the understanding of continuous functions on a closed interval, we may conclude that the maximum of such a function can indeed occur at more than one value within the interval. Hence the statement is True.