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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

Advanced Engineering Mathematics
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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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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