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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.

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.

Calculus For The Life Sciences
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
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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) ≤tf(x₁) + (1 -t) f(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 assignment.
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) ≤tf(x₁) + (1 -t) f(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 assignment.
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