- #1

- 584

- 0

1. A percel delivery service requires that the dimension of a rectangular box be such that the length plus twice the width plus twice the height be no more than 108 centimetres. What is the volume of the largest box that the company will deliver?

2. Let [itex]A = \left( {a_{ij} } \right)[/itex] be a symmetric n * n matrix (ie. a_ij = a_ji) and define

[tex]f:R^n \to R,f\left( {\mathop x\limits^ \to } \right) = \mathop x\limits^ \to \bullet A\mathop x\limits^ \to [/tex]

Suppose x_0 (a point in R^n) is a point where f has a maximum or minimum on the unit sphere [itex]\left\{ {\mathop x\limits^ \to \in R^n :\mathop x\limits^ \to \bullet \mathop x\limits^ \to = 1} \right\}[/itex].

Use Lagrange multipliers to show that x_0 is an eigenvector of A with eigenvalue [tex]\lambda = f\left( {\mathop {x_0 }\limits^ \to } \right)[/tex].

1. In the first one I use Lagrange multipliers and keep on getting roughly V = (27/2)(27)(27/2) ~ 4920 cubic centimetres. The answer is 11664. I basically tried to find max(V = xyz) subject to the contraint P = y + 2x + 2z. (where p <= 108)

An alternative to Lagrange multipliers I guess would be to solve for z in terms of x and y and sub into v = xyz. But looking at the equation with P, if I used the substitution method I would get different answers depending on whatever I solved for y in terms of the other variables or x in terms of the other variables).

The equations which come up when I use Lagrange multipliers are really simple so I don't see why I keep on getting the wrong answer. So I suspect that my constraint expression is incorrect but at the moment I can't spot what's wrong.

2. This one, the hint is to use a result from another question. That is, with f defined as it is, it follows that [tex]\nabla f\left( {\mathop x\limits^ \to } \right) = 2A\mathop x\limits^ \to [/tex].

I don't really see what I can do with it at the moment. I think I'll be dealing with some equations of the form [tex]2a_{k1} + ...2a_{kn} = 2\lambda x_k[/tex]. Nothing comes to mind when I try to do this question. I managed to show the result given in the hint but I'm not sure if knowing how to do so will shed any light on this question.

Basically, I'm tired, out of ideas and I don't know what to do. Can someone please help me out?