3. Determine whether the following subsets of R are bounded above, bounded below,both or neither. Rigorously justify your answers.1(a) A = {x + = | x ≤ (0, ∞)}X(b) B = {x² + xy² | −2 ≤ x ≤ 1, −1 ≤ y ≤ 1}.

Answered: Image /qna-images/answer/8775e23d-cfb7-4143-8ed3-adf103966c85.jpg

Answered: 3. Determine whether the following…

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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10. Prove the following inequaltities for any positive real numbers x,y: @xy² ≤ x² + y²4 (ii) xy³ + x³y ≤x4+y4.
5. Suppose that f(z) is entire and |ƒ(z)| ≤ 2(1 + |z|)4. Show that f(z) is a polynomial of degree at most four.
1. Let DC R. Write down precise mathematical formulations of the following statements, using quantifiers (V, 3): (a) D is bounded above. (b) D has a maximum element. (c) D does not have a maximum element.
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.)
B₂ {(₂1 X12 X22 | Xij ≥ 0 for i, j = 1, 2, X11 + 12 = 2, X21 + x22 = 2, X11 + x21 (6) Let B₂ to be the set described in the screenshot attached Suppose Cij are real numbers such that C11C22 C12 + C21 > 0. Consider the following LPP: maximize: C11x11 + C12x12 + C21%21+ C22X22 subject to: (#11 #12) € B₂ X21 = 1.5, x12 + x22 = 2. (a) Write this problem in canonical form {x = (x11, 12, X21, X22) € R4 | Ax = b, x ≥ 0}. Hint: A should be a 3 x 4 matrix (after you eliminate one row which is linearly dependent on the others). (b) List all BFS. Which BFS x* has the maximum cost for this LPP? 2.5}. For the maximizer x*, show that for all basic directions D₁ at x* the reduced cost Cj = c² Dj ≤ 0.
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3. Determine whether the following subsets of R are bounded above, bounded below, both or neither. Rigorously justify your answers. 1 (a) A = {x + = | x ≤ (0, ∞)} X (b) B = {x² + xy² | −2 ≤ x ≤ 1, −1≤ y ≤ 1}.