yEXAMPLE 5Sketch a possible graph of a function f that satisfies the following conditions.(i)f'(x) > 0 on (-∞, 1), f'(x) < 0 on (1, o)(ii)f"(x) > 0 on (-∞, -2) and (2, 0), f"(x) < 0 on (-2, 2)(ii)lim f(x) = -3, lim f(x) =X → 00X> -0012SOLUTIONCondition (i) tells us that f is increasing on (-o, 1) and decreasing on (1, o). Condition (ii) says that f is concave upward on-00,andand concave downward onFrom (iii) we know that the graph of f has two horizontal asymptotes:00y =and y = 0.Video ExampleFirst we draw the horizontal asymptote y =as a dashed line (see the figure). We then draw the graph of f approaching this asymptote at the farleft, increasing to its maximum point at x = 1 and decreasing toward the x-axis at the far right. We also make sure that the graph has inflection pointsNotice that we made the curve bend upward for x < -2 and x > 2, and bend downward when x is betweenwhen x = -2 andand 2.

Given conditions of function are i.) f'x>0 on -∞, 1 and f'x<0 on 1, ∞. i.e. function is…

EXAMPLE 5 Sketch a possible graph of a function f that satisfies the following conditions. (1) r(x) > 0 on (-», 1), f'(x) < 0 on (1, ) (ii) f"(x) > 0 on (-», -2) and (2, ), f"(x) < 0 on (-2, 2) (iii) lim f(x) = -3, lim f(x) = 0

EXAMPLE 5 Sketch a possible graph of a function f that satisfies the following conditions. (1) r(x) > 0 on (-», 1), f'(x) < 0 on (1, ) (ii) f"(x) > 0 on (-», -2) and (2, ), f"(x) < 0 on (-2, 2) (iii) lim f(x) = -3, lim f(x) = 0

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)

6th Edition

ISBN:9781337111348

Author:Bruce Crauder, Benny Evans, Alan Noell

Publisher:Bruce Crauder, Benny Evans, Alan Noell

Chapter5: A Survey Of Other Common Functions

Section5.6: Higher-degree Polynomials And Rational Functions

Problem 14E

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Concept explainers

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.

Question

y
EXAMPLE 5
Sketch a possible graph of a function f that satisfies the following conditions.
(i)
f'(x) > 0 on (-∞, 1), f'(x) < 0 on (1, o)
(ii)
f"(x) > 0 on (-∞, -2) and (2, 0), f"(x) < 0 on (-2, 2)
(ii)
lim f(x) = -3, lim f(x) =
X → 00
X> -00
1
2
SOLUTION
Condition (i) tells us that f is increasing on (-o, 1) and decreasing on (1, o). Condition (ii) says that f is concave upward on
-00,
and
and concave downward on
From (iii) we know that the graph of f has two horizontal asymptotes:
00
y =
and y = 0.
Video Example
First we draw the horizontal asymptote y =
as a dashed line (see the figure). We then draw the graph of f approaching this asymptote at the far
left, increasing to its maximum point at x = 1 and decreasing toward the x-axis at the far right. We also make sure that the graph has inflection points
Notice that we made the curve bend upward for x < -2 and x > 2, and bend downward when x is between
when x = -2 and
and 2.

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