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Chapter 3: Q. 4 (page 247)
If f has a local minimum at x = 0 and a local maximum at x = 2, what can you say about f '(0) and f '(2)? Is there anything else you can say about f ' .
If f has local maximum atand local minimum atthenmust be equal to zero .
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Sketch careful, labeled graphs of each function f in Exercises 63–82 by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of and examine any relevant limits so that you can describe all key points and behaviors of f.
Determine whether or not each function f in Exercises 53–60 satisfies the hypotheses of the Mean Value Theorem on the given interval [a, b]. For those that do, use derivatives and algebra to find the exact values of all c ∈ (a, b) that satisfy the conclusion of the Mean Value Theorem.
Use a sign chart for to determine the intervals on which each function is increasing or decreasing. Then verify your algebraic answers with graphs from a calculator or graphing utility.
For each set of sign charts in Exercises 53–62, sketch a possible graph of f.
Use the definitions of increasing and decreasing to argue that is decreasing on and increasing on . Then use derivatives to argue the same thing.
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