(a) Decide whether the following statements are true; briefly justify your an- swers (you may quote any theorem from the lectures). (i) Every continuous function f : [-9, 11] → R has a minimum. (ii) Every continuous function f : (-5, 1] → R has a maximum. (iii) Every continuous function f : (−1, 1] → R has a supremum. (iv) The set [0,8) has a supremum but no maximum. (b) Let f R R be a differentiable function. Let a, b = R be such that a
(a) Decide whether the following statements are true; briefly justify your an- swers (you may quote any theorem from the lectures). (i) Every continuous function f : [-9, 11] → R has a minimum. (ii) Every continuous function f : (-5, 1] → R has a maximum. (iii) Every continuous function f : (−1, 1] → R has a supremum. (iv) The set [0,8) has a supremum but no maximum. (b) Let f R R be a differentiable function. Let a, b = R be such that a
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![(a) Decide whether the following statements are true; briefly justify your an-
swers (you may quote any theorem from the lectures).
(i) Every continuous function f : [-9, 11] → R has a minimum.
(ii) Every continuous function f : (-5, 1] → R has a maximum.
(iii) Every continuous function f : (−1, 1] → R has a supremum.
(iv) The set [0,8) has a supremum but no maximum.
(b) Let f R R be a differentiable function. Let a, b = R be such that
a <band f(a) = a and f(b) = b. Show that there exists c = (a, b) such
that f'(c) 1. You may use any theorem proved in the lectures.
=
(c) Use the Intermediate Value Theorem and Rolle's Theorem to show that
the function g(x) = 2x3 + 3x + 4 has exactly one real root.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F017b5c7e-49c0-46c7-a22b-9336d465c7c9%2F1faa9b7c-e919-4230-a02e-bd32e77ef810%2F9sfiu5w_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(a) Decide whether the following statements are true; briefly justify your an-
swers (you may quote any theorem from the lectures).
(i) Every continuous function f : [-9, 11] → R has a minimum.
(ii) Every continuous function f : (-5, 1] → R has a maximum.
(iii) Every continuous function f : (−1, 1] → R has a supremum.
(iv) The set [0,8) has a supremum but no maximum.
(b) Let f R R be a differentiable function. Let a, b = R be such that
a <band f(a) = a and f(b) = b. Show that there exists c = (a, b) such
that f'(c) 1. You may use any theorem proved in the lectures.
=
(c) Use the Intermediate Value Theorem and Rolle's Theorem to show that
the function g(x) = 2x3 + 3x + 4 has exactly one real root.
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