x2 (Figure 24). Verify the following: 1) Let f(x) (a) f(0) is a local max and f(2) a local min. (b) ƒ is concave down on (-∞, 1) and concave up on (1, ∞0). (c) lim f(x) = -o and lim f(x) = ∞. x-1- x→1+ (d) y = x + 1 is a slant asymptote of f(x) as x → ±∞. (e) The slant asymptote lies above the graph of f(x) for x < 1 and below the graph for x > 1.
x2 (Figure 24). Verify the following: 1) Let f(x) (a) f(0) is a local max and f(2) a local min. (b) ƒ is concave down on (-∞, 1) and concave up on (1, ∞0). (c) lim f(x) = -o and lim f(x) = ∞. x-1- x→1+ (d) y = x + 1 is a slant asymptote of f(x) as x → ±∞. (e) The slant asymptote lies above the graph of f(x) for x < 1 and below the graph for x > 1.
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter3: Functions
Section3.3: More On Functions; Piecewise-defined Functions
Problem 99E: Determine if the statemment is true or false. If the statement is false, then correct it and make it...
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11/30,
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![1) Let f(x)
(Figure 24). Verify the following:
X – 1
(a) f(0) is a local max and f(2) a local min.
(b) f is concave down on (-o, 1) and concave up on (1, ∞).
(c) lim f(x) = -∞ and lim f(x) = 0o.
x→1-
x→1+
(d) y = x + 1 is a slant asymptote of f(x) as x → ±0.
(e) The slant asymptote lies above the graph of f(x) for x < 1 and below the graph for x > 1.
y
f(x) =
10
y =x + 1
-10
10
-10
FIGURE 24
2) Let f be continuous on [a, b], differentiable on (a, b) and positive (i.e., > 0) for all x E (a, b). Prove that there
exists c E (a, b) such that/ (C)
f(c)
+ (Hint: consider the function F (x) = (x – a) (x – b) ƒ (x) and use
b-c
MVT for F (x) to show the existence of such c E (a, b).)
3) Use L'Hopital's Rule to evaluate and check your answers numerically:
sin x
(a) lim
x→0+
1
(b) lim
sin? x
x2
x→0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7fd7061d-729c-4718-a2e7-fb53d4373f92%2F8f8d495e-bf8a-4301-8edc-7f07a954352e%2Fh3oepxd_processed.png&w=3840&q=75)
Transcribed Image Text:1) Let f(x)
(Figure 24). Verify the following:
X – 1
(a) f(0) is a local max and f(2) a local min.
(b) f is concave down on (-o, 1) and concave up on (1, ∞).
(c) lim f(x) = -∞ and lim f(x) = 0o.
x→1-
x→1+
(d) y = x + 1 is a slant asymptote of f(x) as x → ±0.
(e) The slant asymptote lies above the graph of f(x) for x < 1 and below the graph for x > 1.
y
f(x) =
10
y =x + 1
-10
10
-10
FIGURE 24
2) Let f be continuous on [a, b], differentiable on (a, b) and positive (i.e., > 0) for all x E (a, b). Prove that there
exists c E (a, b) such that/ (C)
f(c)
+ (Hint: consider the function F (x) = (x – a) (x – b) ƒ (x) and use
b-c
MVT for F (x) to show the existence of such c E (a, b).)
3) Use L'Hopital's Rule to evaluate and check your answers numerically:
sin x
(a) lim
x→0+
1
(b) lim
sin? x
x2
x→0
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