Suppose that a function f is continuous on the closed interval [0,1] and that 0sf(x)s1 for every x in [0,1]. Show that there must exist a number c on [0,1] such that f(c) = c (c is called a fixed point of f). In which scenario is the proof trivial? O A. f(0) = 1 or f(1) = 0 O B. f(0) = 1 and f(1) = 0 OC. f(0) = 0 or f(1) = 1 Let f(0) = a and f(1) = b. If the scenario above is excluded from consideration, what can safely be assumed about the values of a and b? O A. a<1 and b>0 O B. a>0 and b<1 O C. a<0 and b>1 O D. a>1 and b<0 Define g(x) = f(x) - x. Is g(x) continuous on [0,1]? O Yes O No Use the inequalities for the values of a and b to find new inequalities limiting the values of g(0) and g(1). Choose the correct answer below. O A. g(0) < 0 and g(1) < 0 O B. g(0) > 0 and g(1) < 0 O C. g(0) > 0 and g(1) > 0 O D. g(0) < 0 and g(1) > 0 By the Intermediate Value Theorem and the inequalities above, there must be some c in [0,1] such that g(c) =| Solve for f(x) in the equation g(x) = f(x) – x and evaluate at x = c using the value determined for g(c). f(c) =|

Suppose that a function f is continuous on the closed interval [0,1] and that 0sf(x)s1 for every x in [0,1]. Show that there must exist a number c on [0,1] such that f(c) = c (c is called a fixed point of f). In which scenario is the proof trivial? O A. f(0) = 1 or f(1) = 0 O B. f(0) = 1 and f(1) = 0 OC. f(0) = 0 or f(1) = 1 Let f(0) = a and f(1) = b. If the scenario above is excluded from consideration, what can safely be assumed about the values of a and b? O A. a<1 and b>0 O B. a>0 and b<1 O C. a<0 and b>1 O D. a>1 and b<0 Define g(x) = f(x) - x. Is g(x) continuous on [0,1]? O Yes O No Use the inequalities for the values of a and b to find new inequalities limiting the values of g(0) and g(1). Choose the correct answer below. O A. g(0) < 0 and g(1) < 0 O B. g(0) > 0 and g(1) < 0 O C. g(0) > 0 and g(1) > 0 O D. g(0) < 0 and g(1) > 0 By the Intermediate Value Theorem and the inequalities above, there must be some c in [0,1] such that g(c) =| Solve for f(x) in the equation g(x) = f(x) – x and evaluate at x = c using the value determined for g(c). f(c) =|

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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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

Suppose that a function f is continuous on the closed interval [0,1] and that 0s (x)s1 for every x in [0,1]. Show that there must exist a number c on [0,1] such that f(c) = c (c is called a fixed point of f).
In which scenario is the proof trivial?
O A. f(0) = 1 or f(1) = 0
O B. f(0) = 1 and f(1) = 0
OC. (0) = 0 or f(1) = 1
Let f(0) = a and f(1) = b. If the scenario above is excluded from consideration, what can safely be assumed about the values of a and b?
O A. a<1 and b>0
O B. a>0 and b<1
O C. a<0 and b>1
O D. a>1 and b<0
Define g(x) = f(x) - x. Is g(x) continuous on [0,1]?
O Yes
O No
Use the inequalities for the values of a and b to find new inequalities limiting the values of g(0) and g(1). Choose the correct answer below.
O A. g(0) < 0 and g(1) < 0
O B. g(0) >0 and g(1) < 0
O C. g(0) > 0 and g(1) > 0
O D. g(0) <0 and g(1) >0
By the Intermediate Value Theorem and the inequalities above, there must be some c in [0,1] such that g(c) =
Solve for f(x) in the equation g(x) = f(x) – x and evaluate at x= c using the value determined for g(c.
f(c) =O

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