$3-4$ . For each of the numbers $a, b, c, d, r,$ …

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Problem 2

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?

Answer

A.Extreme Value Theorem B. Step $1 :$ Find all values of $x$ for which $f^{\prime}(x)=0$ or $f^{\prime}(x)$ is undefined. Step $2 :$ Evaluate $f(x)$ at those values of $x$ as well as at the endpoints. Step $3 :$ Compare these finitely many $f(x)$ to find the largest and smallest value.

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