Concept explainers
a.
To find: the absolute extrema of f and where they occur.
a.
Answer to Problem 52E
The absolute maximum is
Explanation of Solution
Given information: f is an even function, continuous
x | 0 | 1 | 2 |
f | 2 | 0 | −1 |
f’ | Does not exist | 0 | Does not exist |
f’’ | Does not exist | 0 | Does not exist |
x | 0 | 1 | 2 |
f | + | − | − |
f’ | − | − | + |
f’’ | + | − | − |
f is an even function; therefore, the table can be extended for the values of x between −3 and 0. Since it is even the portion on the left will be the mirror image of the portion on the right, that is, the y -coordinates remain same, the slopes have opposite sign and the concavity remains same.
x | −2 | −1 | 0 | 1 | 2 |
f | −1 | 0 | 2 | 0 | −1 |
f’ | Does not exist | 0 | Does not exist | 0 | Does not exist |
f’’ | Does not exist | 0 | Does not exist | 0 | Does not exist |
x | −3 | −2 | −1 | 0 | 1 | 2 |
f | − | − | + | + | − | − |
f’ | − | + | + | − | − | + |
f’’ | − | − | + | + | − | − |
Since, f < 0 for −3 < x < −2 and 2
Since, f < 0 for −3
Hence, the absolute maximum is
b.
To find: the points of inflection.
b.
Answer to Problem 52E
The points of inflection are
Explanation of Solution
Given information: f is an even function, continuous
X | 0 | 1 | 2 |
F | 2 | 0 | −1 |
f’ | Does not exist | 0 | Does not exist |
f’’ | Does not exist | 0 | Does not exist |
X | 0 | 1 | 2 |
F | + | − | − |
f’ | − | − | + |
f’’ | + | − | − |
Hence, the points of inflection are
c.
To graph: the function f with the provided information.
c.
Explanation of Solution
Given information: f is an even function, continuous
X | 0 | 1 | 2 |
F | 2 | 0 | −1 |
f’ | Does not exist | 0 | Does not exist |
f’’ | Does not exist | 0 | Does not exist |
X | 0 | 1 | 2 |
F | + | − | − |
f’ | − | − | + |
f’’ | + | − | − |
Graph:
Interpretation:
Hence, this is the probable graph for function f with the provided information.
Chapter 5 Solutions
Calculus: Graphical, Numerical, Algebraic: Solutions Manual
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