3) Extrema (a) Let f : {x Є R | x > −5 ^ x + 1} → R with 1 In f(z) = ± 1 + (3 + 1). x- Determine all elements of the domain where f assumes local extrema. Specify in each case whether it is a local maximum or minimum. (b) Let f : [a, b] → R be an at least twice continuously differentiable, convex function, which has a local minimum at x € (a, b). Prove that there can be no x1 € (a,b) with f(x1) < f(x) (the minimum is therefore even a global minimum). Tips: Monotonicity of the first derivative, mean value theorem.
3) Extrema (a) Let f : {x Є R | x > −5 ^ x + 1} → R with 1 In f(z) = ± 1 + (3 + 1). x- Determine all elements of the domain where f assumes local extrema. Specify in each case whether it is a local maximum or minimum. (b) Let f : [a, b] → R be an at least twice continuously differentiable, convex function, which has a local minimum at x € (a, b). Prove that there can be no x1 € (a,b) with f(x1) < f(x) (the minimum is therefore even a global minimum). Tips: Monotonicity of the first derivative, mean value theorem.
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
![3) Extrema
(a) Let f : {x Є R | x > −5 ^ x + 1} → R with
1
In
f(z) = ± 1 + (3 + 1).
x-
Determine all elements of the domain where f assumes local extrema. Specify in each case whether
it is a local maximum or minimum.
(b) Let f : [a, b] → R be an at least twice continuously differentiable, convex function, which has a
local minimum at x € (a, b). Prove that there can be no x1 € (a,b) with f(x1) < f(x) (the
minimum is therefore even a global minimum).
Tips: Monotonicity of the first derivative, mean value theorem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9fe7e31c-e634-45b0-ad0d-23924a70fec5%2F34b5b655-e03d-4fbd-a1f3-2311e075e0e4%2Fgct402s_processed.png&w=3840&q=75)
Transcribed Image Text:3) Extrema
(a) Let f : {x Є R | x > −5 ^ x + 1} → R with
1
In
f(z) = ± 1 + (3 + 1).
x-
Determine all elements of the domain where f assumes local extrema. Specify in each case whether
it is a local maximum or minimum.
(b) Let f : [a, b] → R be an at least twice continuously differentiable, convex function, which has a
local minimum at x € (a, b). Prove that there can be no x1 € (a,b) with f(x1) < f(x) (the
minimum is therefore even a global minimum).
Tips: Monotonicity of the first derivative, mean value theorem.
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