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Solution set of a quadratic inequality. Let CCR" be the solution set of a quadratic inequality, C = {r € R¹ | x¹ Ax+bx+c <0}, with A E S", bER", and c E R. (a) Show that C is convex if A 0. (b) Show that the intersection of C and the hyperplane defined by ga+h=0 (where g 0) is convex if A+ Agg¹0 for some À E R. Are the converses of these statements true?

Solution set of a quadratic inequality. Let CCR" be the solution set of a quadratic inequality, C = {r € R¹ | x¹ Ax+bx+c <0}, with A E S", bER", and c E R. (a) Show that C is convex if A 0. (b) Show that the intersection of C and the hyperplane defined by ga+h=0 (where g 0) is convex if A+ Agg¹0 for some À E R. Are the converses of these statements true?

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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Solution set of a quadratic inequality. Let C C R" be the solution set of a quadratic inequality, C = {r € R" | x" Ax + b"x +c< 0}, with A E S", be R", and c e R. (a) Show that C is convex if A > 0. (b) Show that the intersection of C and the hyperplane defined by g"x + h = 0 (where g + 0) is convex if A+ Agg' = 0 for some X e R. Are the converses of these statements true?
Transcribed Image Text:Solution set of a quadratic inequality. Let C C R" be the solution set of a quadratic inequality, C = {r € R" | x" Ax + b"x +c< 0}, with A E S", be R", and c e R. (a) Show that C is convex if A > 0. (b) Show that the intersection of C and the hyperplane defined by g"x + h = 0 (where g + 0) is convex if A+ Agg' = 0 for some X e R. Are the converses of these statements true?
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