Question

True or False A continuous function on a closed interval must attain a maximum value on that interval. Justify your answer.

Short answer

True. According to the Extreme Value Theorem, any continuous function on a closed interval must attain a maximum value on that interval.

Step-by-step solution

State the Extreme Value Theorem

The Extreme Value Theorem states that if a function \(f(x)\) is continuous on a closed interval \([a, b]\), then \(f(x)\) attains a maximum and a minimum value on the interval \([a, b]\). This theorem implies that there exists numbers \(c\) and \(d\) in \([a, b]\) such that \(f(c) \leq f(x) \leq f(d)\) for all \(x\) in \([a, b]\).

Relate Theorem to the Exercise

The question in this exercise is if a continuous function on a closed interval must attain a maximum value on that interval. According to the Extreme Value Theorem, this statement is 'True'. Any continuous function on a closed interval must attain both a maximum and minimum value on that interval.

Justify the Answer

The justification for the answer is directly from the Extreme Value Theorem. The conditions of the theorem match the conditions in the question (a continuous function on a closed interval), and the theorem directly states that a maximum value must be attained in these conditions.