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1. A rectangular box is placed in the first octant, with one corner at the origin and the three adjacent faces in the coordinate planes. The opposite point P(r, y, 2) is constrained to lie on the paraboloid r2 + y? + z = 1. Which P gives the box of greatest volume? (a) Show that this problem leads to maximizing f(r, y, 2) = ry – °y – ry³. %3D (b) Find the critical point of f in the first octant. (c) Identity the nature of the critical point using the second derivative test and find the maximum of f in the first octant.

1. A rectangular box is placed in the first octant, with one corner at the origin and the three adjacent faces in the coordinate planes. The opposite point P(r, y, 2) is constrained to lie on the paraboloid r2 + y? + z = 1. Which P gives the box of greatest volume? (a) Show that this problem leads to maximizing f(r, y, 2) = ry – °y – ry³. %3D (b) Find the critical point of f in the first octant. (c) Identity the nature of the critical point using the second derivative test and find the maximum of f in the first octant.

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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1. A rectangular box is placed in the first octant, with one corner at the origin and the three adjacent faces in the coordinate planes. The opposite point P(r, y, z) is constrained to lie on the paraboloid r2 + y² + z = 1. Which P gives the box of greatest volume? (a) Show that this problem leads to maximizing f(r, y, z) = ry – r³y – ry³. %3D (b) Find the critical point of f in the first octant. (c) Identity the nature of the critical point using the second derivative test and find the maximum of ƒ in the first octant.
Transcribed Image Text:1. A rectangular box is placed in the first octant, with one corner at the origin and the three adjacent faces in the coordinate planes. The opposite point P(r, y, z) is constrained to lie on the paraboloid r2 + y² + z = 1. Which P gives the box of greatest volume? (a) Show that this problem leads to maximizing f(r, y, z) = ry – r³y – ry³. %3D (b) Find the critical point of f in the first octant. (c) Identity the nature of the critical point using the second derivative test and find the maximum of ƒ in the first octant.
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