   Chapter 3.1, Problem 14E

Chapter
Section
Textbook Problem

# (a) Sketch the graph of a function that has two local maxima, one local minimum, and no absolute minimum.(b) Sketch the graph of a function that has three local minima, two local maxima, and seven critical numbers.

To determine

Part (a):

To sketch:

The graph of function that has two local maxima, one local minimum, and no absolute minimum

Explanation

1) Concept:

Use the concept of absolute minimum, local maximum and minimum, and extreme value theorem to sketch the graph of a function.

2) Definition:

i. Let c be a number in the domain D of a function f. Then f(c) is the absolute minimum value of f on D if fcf(x) for all x in D.

ii. The number f(c) is a local maximum value of f  if fcf(x) when x is near c and local minimum value of f if fc fx  when x is near c.

Extreme value theorem:

If f is continuous on a closed interval a, b, then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers c and d in [a, b].

4) Given:

The function has two local maxima, one local minimum, and no absolute minimum

To determine

Part (b):

To sketch:

The graph of function that has two local maxima, three local minima, and seven critical numbers.

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