Suppose f is differentiable and f' is increasing on an interval (a, b) (that is, f is concave-u according to Definition 4.6.1). Prove that f should be also convex.
Suppose f is differentiable and f' is increasing on an interval (a, b) (that is, f is concave-u according to Definition 4.6.1). Prove that f should be also convex.
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter9: Multivariable Calculus
Section9.3: Maxima And Minima
Problem 29E
Related questions
Question
![3. We have learned the definitions of concavity (up and down) for differentiable functions
following Definition 4.6.1 in the textbook. In this question, we learn a new definition of
concave-up (also known as convex) that applies to any function as long as the function is
defined on an interval, not necessarily differentiable.
An important question, then, would be " are these two definitions equivalent to each other
for differentiable functions?" You are going to show that it is partially true.
Let f be a function defined on an interval (a, b).
We say that a function f is convex if for every a < x₁ < x2 < b and every 0 < t < 1 the
following inequality holds
f(tx₁ + (1 − t)x₂) ≤ tƒ (x₁) + (1 − t) ƒ (x₂)
In other words, f is convex if the line segment between any two distinct points on the
graph of the function lies above the graph between the two points.
Suppose f is differentiable and f' is increasing on an interval (a, b) (that is, f is concave-up
according to Definition 4.6.1). Prove that f should be also convex.
Notes: You may wonder if the converse is also true. For the converse to be true we need
a weaker definition of "increasing". This is not part of this](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F64e22453-08e9-4901-b6d4-29904b54e3b3%2F6a1c69d1-f3af-4bde-a32d-28ba32f37301%2Fowgsyh_processed.png&w=3840&q=75)
Transcribed Image Text:3. We have learned the definitions of concavity (up and down) for differentiable functions
following Definition 4.6.1 in the textbook. In this question, we learn a new definition of
concave-up (also known as convex) that applies to any function as long as the function is
defined on an interval, not necessarily differentiable.
An important question, then, would be " are these two definitions equivalent to each other
for differentiable functions?" You are going to show that it is partially true.
Let f be a function defined on an interval (a, b).
We say that a function f is convex if for every a < x₁ < x2 < b and every 0 < t < 1 the
following inequality holds
f(tx₁ + (1 − t)x₂) ≤ tƒ (x₁) + (1 − t) ƒ (x₂)
In other words, f is convex if the line segment between any two distinct points on the
graph of the function lies above the graph between the two points.
Suppose f is differentiable and f' is increasing on an interval (a, b) (that is, f is concave-up
according to Definition 4.6.1). Prove that f should be also convex.
Notes: You may wonder if the converse is also true. For the converse to be true we need
a weaker definition of "increasing". This is not part of this
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 23 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Recommended textbooks for you
![Calculus For The Life Sciences](https://www.bartleby.com/isbn_cover_images/9780321964038/9780321964038_smallCoverImage.gif)
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
![Calculus For The Life Sciences](https://www.bartleby.com/isbn_cover_images/9780321964038/9780321964038_smallCoverImage.gif)
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage