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Exercise III.4. For any r R and function f(): R → R, recall (see II.7) the r-level set is defined by r-levf := {x: f(x) ≤r}. Show that if f(-) is convex, then r-lev, € C. Give an example where each r-lev, is convex, but f() is not a convex function. FOR REFERENCE Exercise II.7. Find all local max and min's for the following functions (n = 2). (a) fo (1) = x³ - I. (b) fio (4) = y + (x − 1)² + (y + 2)². (c) fu (4) = xy + (x + 2)² + (y-2)². (d) f12 (²) = (2+2)²- (x+2)² + (y-3)² = 5x + 6y. 9

Exercise III.4. For any r R and function f(): R → R, recall (see II.7) the r-level set is defined by r-levf := {x: f(x) ≤r}. Show that if f(-) is convex, then r-lev, € C. Give an example where each r-lev, is convex, but f() is not a convex function. FOR REFERENCE Exercise II.7. Find all local max and min's for the following functions (n = 2). (a) fo (1) = x³ - I. (b) fio (4) = y + (x − 1)² + (y + 2)². (c) fu (4) = xy + (x + 2)² + (y-2)². (d) f12 (²) = (2+2)²- (x+2)² + (y-3)² = 5x + 6y. 9

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.4: Definition Of Function
Problem 64E
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Exercise III.4. For any r R and function f(): R → R, recall (see II.7) the r-level set is defined by r-levƒ := {x: f(x) ≤r}. Show that if f() is convex, then r-lev, € C. Give an example where each r-lev, is convex, but f() is not a convex function. FOR REFERENCE Exercise II.7. Find all local max and min's for the following functions (n = 2). (a) fo (1) = x³ - I. (b) fio (1) = y + (x − 1)² + (y + 2)². (c) fu (4) = xy + (x + 2)² + (y-2)². (d) f12 (~) = (²+2)² + (x-3)² – 5x + 6y. 9
Transcribed Image Text:Exercise III.4. For any r R and function f(): R → R, recall (see II.7) the r-level set is defined by r-levƒ := {x: f(x) ≤r}. Show that if f() is convex, then r-lev, € C. Give an example where each r-lev, is convex, but f() is not a convex function. FOR REFERENCE Exercise II.7. Find all local max and min's for the following functions (n = 2). (a) fo (1) = x³ - I. (b) fio (1) = y + (x − 1)² + (y + 2)². (c) fu (4) = xy + (x + 2)² + (y-2)². (d) f12 (~) = (²+2)² + (x-3)² – 5x + 6y. 9
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