(b) Show that definition (2) is equivalent to definition (3). Hint: You may need to use the Mean Value Theorem. 1. There are several equivalent definitions of concavity. (1) We say a twice-differentiable function f to be (strictly) concave up on (a, b) if for any distinct T1, T2 (a, b)(assume x1 < x2) and for any t = (0, 1) we have f(x1+t(x2 − x1)) < f(x1)+t(f(x2) − f(x1)). (2) We say a twice-differentiable function f to be (strictly) concave up on (a, b) if for any distinct T1, T2 € (a, b)(assume x1 < x2) and for any 1 € (11, 12), f(x) f(x1) f(x2) f(x1) f(x2)- f(x) x-x1 < I2X1 < X2-X (3) We say a twice-differentiable function f to be (strictly) concave up on (a,b) if f' is (strictly) increasing on (a,b).
(b) Show that definition (2) is equivalent to definition (3). Hint: You may need to use the Mean Value Theorem. 1. There are several equivalent definitions of concavity. (1) We say a twice-differentiable function f to be (strictly) concave up on (a, b) if for any distinct T1, T2 (a, b)(assume x1 < x2) and for any t = (0, 1) we have f(x1+t(x2 − x1)) < f(x1)+t(f(x2) − f(x1)). (2) We say a twice-differentiable function f to be (strictly) concave up on (a, b) if for any distinct T1, T2 € (a, b)(assume x1 < x2) and for any 1 € (11, 12), f(x) f(x1) f(x2) f(x1) f(x2)- f(x) x-x1 < I2X1 < X2-X (3) We say a twice-differentiable function f to be (strictly) concave up on (a,b) if f' is (strictly) increasing on (a,b).
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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Please answer part b of this question. May I ask for a handwritten and non-AI-generated answer to this question since I really need a well-rounded step and explanation to understand the materials? Thank you!
Transcribed Image Text:(b) Show that definition (2) is equivalent to definition (3).
Hint: You may need to use the Mean Value Theorem.
Transcribed Image Text:1. There are several equivalent definitions of concavity.
(1) We say a twice-differentiable function f to be (strictly) concave up on (a, b) if for any distinct
T1, T2 (a, b)(assume x1 < x2) and for any t = (0, 1) we have
f(x1+t(x2 − x1)) < f(x1)+t(f(x2) − f(x1)).
(2) We say a twice-differentiable function f to be (strictly) concave up on (a, b) if for any distinct
T1, T2 € (a, b)(assume x1 < x2) and for any 1 € (11, 12),
f(x) f(x1) f(x2) f(x1) f(x2)- f(x)
x-x1
<
I2X1
<
X2-X
(3) We say a twice-differentiable function f to be (strictly) concave up on (a,b) if f' is (strictly)
increasing on (a,b).
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