1. For each of the following sets, select all that are convex:(a) S = {(x1, x2) € R² | x² + x² ≤ 4, x1 ≥ 0, x2 ≤ 0}.(b) S = : {(x₁, x₂) = Z²|x² + x² ≤ 4, x₁ ≥ 0, x2 ≤ 0}.(c) S = {x € R¹| Σ1 x² ≤ 4, x₁ ≥ 0, x2 ≤ 0}.(d) S = {x = R¹ |x₁ + x2 = 10(x3 + x4)}, where n ≥ 4.(e) S = {x ER | maxi=1,...,nnhi10 mini 1,..., {i}}.=

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Answered: 1. For each of the following sets,…

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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2. Consider the following sets. Determine which one(s) are convex and which one(s) are not. For any set that is not convex, give two points that show this set is not convex. For any set that is convex, explain why using one of the rules seen in class or using first principles: (a) S₁ = {x € R" | x₁ ≤i, Vi= 1,...,n}. (b) S₂ = {x ER | a? > i, Vi= 1,...,n}. (c) $g= { € R2 \x1+≤2, i+ <4}. (d) S₁ = {x € Z² | x² + x² ≤ 4}. (e) S5 = {x R² | 1₂ ≥ e¹¹}. (f) S6 = {x € R² | x₂ ≤e²²}.
10. Prove the following inequaltities for any positive real numbers x,y: @xy² ≤ x² + y²4 (ii) xy³ + x³y ≤x4+y4.
6. By drawing diagrams, determine which of the following sets is convex. {(x, y): y=e*}; {(x,y):yze}; (x, y): xy ≥ 1, x>0, y > 0.
3. Express the following regions in the stated coordinate systems. You do not need to justify your answer but showing your though process may help with partial marks. Your answer should be a set in the following form: {(V1, V2, V3) : a ≤ v₁ ≤ b, f(v1) ≤ v2 ≤ g(v1), Ø(V1, V2) ≤ v3 ≤ (v₁, v₂)} where (v₁, V2, V3) are the three variables of the coordinate system in questions in any order, a, b R are constants, f, g are functions of v₁ and 6, are functions of v₁, V2. (a) The region outside the cylinder x² + y² ≤ r² and inside the sphere x² + y² + z² r< R. Express this in the cylindrical and spherical coordinate systems. - R² assuming (b) The region in the first octant (x, y, z ≥ 0) below the plane x + 2y + 3z = 6. Express this in the cylindrical, spherical and Cartesian coordinate system.
For each of the following sets, find the interior points, the set of all limit points, isolated points, and boundary points.
8. What is the domain and range of the following relation? y=x−2x−3 y ICI HI -6 -5 2 a) {x ER}, {y ER, y> -4} b) {x E R}, {y E R, y > 0} b) {x ER, x < -4}, {y E R} d) {x ER, x < -3}, {y ER} F
Suppose X∼t(10)X∼t(10). Which of the following code gives you the central limits aa and bb such that P(a<X<b)=0.95P(a<X<b)=0.95? A. a <- qt(0.025, df = 10)b <- qt(0.025, df = 10, lower.tail = F) B. a <- qt(0.975, df = 10, lower.tail = F)b <- qt(0.975, df = 10) C. a <- qt(0.025, df = 10 , lower.tail = F)b <- qt(0.025, df = 10) D. a <- qt(0.975, df = 10)b <- qt(0.975, df = 10, lower.tail = F) E. A and B F. A and D G. B and C H. C and D
Answer ONLY part C
1.54. (!) Let S = {(x, y) e R²: y ≤ x and x + 3y ≥ 8 and x ≤ 8). a) Graph the set S. b) Find the minimum value of x + y such that (x, y) e S. (Hint: On the graph from part (a), sketch the level sets of the function f defined by f(x, y) = x + y.)
-2, r<0, 9. Let f: R R be defined by f(x) then f-(]1,3[) (the inverse image of the r+ 2, x20 interval ]1,3[ under f) is (a) ]3, 5[U] – 1,1[ (b) [0, 1[ (c) 1- 1, 1[ (d) None of the above
Which of the following pairs of sets are equivalent or equal?

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1. For each of the following sets, select all that are convex: (a) S = {(1, ₂) € R² | x² + x² ≤ 4, x₁ ≥ 0, x₂ ≤ 0}. (b) S = {(x1, x2) € ZZ²|x² + x² ≤ 4, x₁ ≥ 0, x2 ≤ 0}. (c) S = {xe R¹|1x² ≤ 4, x₁ ≥ 0,02 ≤ 0}.