What does the Extreme Value Theorem guarantee for a function continuous on a closed interval [a,b]?

What does the Extreme Value Theorem guarantee for a function continuous on a closed interval [a,b]?

• A. It attains a maximum and a minimum on [a,b].
• B. It attains only a maximum value on [a,b].
• C. It attains extrema only at endpoints.
• D. It may not attain any extrema on [a,b].

Answer: (A)

When a function is continuous on a closed interval, the values it takes are kept within a finite range and the endpoints are included, which together prevent the function from “escaping” and ensure it actually hits its extreme values somewhere in that interval. The Extreme Value Theorem guarantees there exist points c and d in [a,b] where f(c) is the largest value and f(d) is the smallest value that f takes on [a,b]. In other words, the function attains both a maximum and a minimum on the interval. The extrema can occur at interior points or at endpoints, and there will always be at least one of each. So the precise guarantee is that it attains both a maximum and a minimum on [a,b].

When a function is continuous on a closed interval, the values it takes are kept within a finite range and the endpoints are included, which together prevent the function from “escaping” and ensure it actually hits its extreme values somewhere in that interval. The Extreme Value Theorem guarantees there exist points c and d in [a,b] where f(c) is the largest value and f(d) is the smallest value that f takes on [a,b]. In other words, the function attains both a maximum and a minimum on the interval. The extrema can occur at interior points or at endpoints, and there will always be at least one of each. So the precise guarantee is that it attains both a maximum and a minimum on [a,b].