Problem 129. Show that if f'(x) < 0 for every x in the interval (a, b) then f is decreasing on (a, b).
Problem 129. Show that if f'(x) < 0 for every x in the interval (a, b) then f is decreasing on (a, b).
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.5: The Kernel And Range Of A Linear Transformation
Problem 30EQ
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Problem 129. Show that if f'(x) < 0 for every x in the interval (a, b) then f
is decreasing on (a, b).
Corollary 4. Suppose f is differentiable on some interval (a, b), f' is con-
tinuous on (a, b), and that f'(c) > 0 for some c e (a, b). Then there is an
interval, I C (a, b), containing c such that for every r, y in I where r 2 y,
f(r) > f(y).
Problem 130. Prove Corollary A
Problem 131. Show that if f is differentiable on some interval (a, b) and that
f'(c) < 0 for some ce (a, b) then there is an interval, I C (a, b), containing c
such that for every x, y in I where r <y, f(x) < f(y).
CONTINUITY: WHAT IT ISN'T AND WHAT IT IS
125
Additional Problems
Problem 132. Use the definition of continuity to prove that the constant func-
tion g(x) = c is continuous at any point a.
Problem 133.
(a) Use the definition of continuity to prove that In x is continuous at 1. [Hint:
You may want to use the fact |In a| <e + -E < In r <e to find a d.]
(b) Use part (a) to prove that Inx is continuous at any positive real number
a. [Hint: In(x) = In(x/a) + In(a). This is a combination of functions
which are continuous at a. Be sure to explain how you know that In(x/a)
is continuous at a.)
Problem 134. Write a formal definition of the statement f is not continuous at
Sx ifx #1
10 ifr = 1
a, and use it to prove that the function f(x) =
is not continuous
at a = 1
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Problem 129. Show that if f'(x) < 0 for every x in the interval (a, b) then f
is decreasing on (a, b).
Corollary 4. Suppose f is differentiable on some interval (a, b), f' is con-
tinuous on (a, b), and that f'(c) > 0 for some c e (a, b). Then there is an
interval, I C (a, b), containing c such that for every r, y in I where r 2 y,
f(r) > f(y).
Problem 130. Prove Corollary A
Problem 131. Show that if f is differentiable on some interval (a, b) and that
f'(c) < 0 for some ce (a, b) then there is an interval, I C (a, b), containing c
such that for every x, y in I where r <y, f(x) < f(y).
CONTINUITY: WHAT IT ISN'T AND WHAT IT IS
125
Additional Problems
Problem 132. Use the definition of continuity to prove that the constant func-
tion g(x) = c is continuous at any point a.
Problem 133.
(a) Use the definition of continuity to prove that In x is continuous at 1. [Hint:
You may want to use the fact |In a| <e + -E < In r <e to find a d.]
(b) Use part (a) to prove that Inx is continuous at any positive real number
a. [Hint: In(x) = In(x/a) + In(a). This is a combination of functions
which are continuous at a. Be sure to explain how you know that In(x/a)
is continuous at a.)
Problem 134. Write a formal definition of the statement f is not continuous at
Sx ifx #1
10 ifr = 1
a, and use it to prove that the function f(x) =
is not continuous
at a = 1
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