Consider the function f(x) x° + x2 - 4 4.x = x +1+ Note that x2 – 4 x2 – 4 22(x2 – 12) (x² – 4)² 8x(x² + 12) (x² – 4)3 f'(x) = and f"(x) = %3D and the x-intercept of the graph of f is approximately r 1.3146. (a) Evaluate all the necessary limits to identify all the linear asymptotes of the graph of f. (b) Construct a table of signs for f' and f" that specifies the following: i. the intervals on which the graph of f is increasing, decreasing, concave up, and concave down, and ii. the values of x at which f has relative extrema, inflection points, and vertical asymptotes (if any). (c) Sketch the graph of f with emphasis on concavity. Label all the intercept points, relative extremum points, inflection points, and holes (if any) with their coordinates. Show the asymptotes of the graph of f and label them with their respective equa- tions.
Consider the function f(x) x° + x2 - 4 4.x = x +1+ Note that x2 – 4 x2 – 4 22(x2 – 12) (x² – 4)² 8x(x² + 12) (x² – 4)3 f'(x) = and f"(x) = %3D and the x-intercept of the graph of f is approximately r 1.3146. (a) Evaluate all the necessary limits to identify all the linear asymptotes of the graph of f. (b) Construct a table of signs for f' and f" that specifies the following: i. the intervals on which the graph of f is increasing, decreasing, concave up, and concave down, and ii. the values of x at which f has relative extrema, inflection points, and vertical asymptotes (if any). (c) Sketch the graph of f with emphasis on concavity. Label all the intercept points, relative extremum points, inflection points, and holes (if any) with their coordinates. Show the asymptotes of the graph of f and label them with their respective equa- tions.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:p³ + x² – 4
4.x
= x +1+
x2
Consider the function f(x)
x² – 4
Note that
4
x²(x² – 12)
(x² – 4)²
8x(x2 + 12)
(x² – 4)³
f'(x) =
and f"(x) =
%3D
-
and the x-intercept of the graph of f is approximately x z 1.3146.
(a) Evaluate all the necessary limits to identify all the linear asymptotes of the graph
of f.
(b) Construct a table of signs for f' and f" that specifies the following:
i. the intervals on which the graph of f is increasing, decreasing, concave up, and
concave down, and
ii. the values of x at which f has relative extrema, inflection points, and vertical
asymptotes (if any).
(c) Sketch the graph of f with emphasis on concavity. Label all the intercept points,
relative extremum points, inflection points, and holes (if any) with their coordinates.
Show the asymptotes of the graph of f and label them with their respective equa-
tions.
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