Question
Suppose $f$ is a continuous function defined on a closed interval $[a, b]$.(a) What theorem guarantees the existence of an absolute maximum value and an absolute minimum value for $f$ ?(b) What steps would you take to find those maximum and minimum values?
Step 1
The Extreme Value Theorem states that if a function \( f \) is continuous on a closed interval \([a, b]\), then \( f \) attains both an absolute maximum and an absolute minimum value on that interval. Show more…
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Suppose $f$ is a continuous function defined on a closed interval $[a, b] .$ $$ \begin{array}{l}{\text { (a) What theorem guarantees the existence of an absolute max- }} \\ {\text { imum value and an absolute minimum value for } f ?} \\ {\text { (b) What steps would you take to find those maximum and }} \\ {\text { minimum values? }}\end{array} $$
Applications of Differentiation
Maximum and Minimum Values
Suppose f is a continuous function defined on a closed interval [1, 3]. (a) Is the existence of an absolute maximum value and an absolute minimum value for f on the interval [1, 3] guaranteed? (b) If no, why not? If yes, what steps would you take to find those maximum and minimum values?
5. Let f(x) be a non-constant, continuous function defined on a closed interval [a, b]. Then by the Extreme Value Theorem, there exist an absolute minimum m and an absolute maximum M. Show that the range of f, {f(x) | a <= x <= b}, is the interval [m, M].
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