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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 + y? +z = 1. Which P gives the box of greatest volume? (a) Show that this problem leads to maximizing f(r, y, z) = ry – ry – ry*. (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, z) 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, z) = ry – ry – ry*. (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.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section: Chapter Questions
Problem 12T
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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, 2) = xy – ay – ry. (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.
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, 2) = xy – ay – ry. (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.
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