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Problem1:

The temp of the circular plate D= {(x1,x2) | x1[tex]^{2}[/tex] + x2[tex]^{2}[/tex] [tex]\leq[/tex] 1} is given by T=2x[tex]^{2}[/tex] -3y[tex]^{2}[/tex] - 2x. Find hottest and coldest points of the plate.

Problem 2

Show that for all (x1,x2,x3) [tex]\in[/tex] R[tex]^{3}[/tex] with x1>0, x2>0, x3>0 and x1x2x3 = 1, we have x1+x2+x3 [tex]\geq[/tex]3

Solution to problem1

First I think there is a typo in the problem. D is given in terms of x1,x2 and T in terms of x,y. Shouldn't they both be in terms of either x1x2 or xy. If so then I have xy value s which I need to plug into T to find the max and min.

Solution to problem2:

This is where I am having real problems. I am not sure what my constraining function is.

What I have done so far is the following (and this is the crucial step which I may have gotten wrong). My question for this problem is at end.

Maximize a1+a2+a3 subject to a1a2a3 = 1. Formulating this gives me the following

i + j + k = [tex]\lambda[/tex](a2a3) i + [tex]\lambda[/tex](a1a3) j + [tex]\lambda[/tex](a1a2) k

Therefore

1 = [tex]\lambda[/tex](a2a3);

1 = [tex]\lambda[/tex](a1a3);

1 = [tex]\lambda[/tex](a1a2);

Multiplying lhs and rhs in above 3 gives me following

a1 = [tex]\lambda[/tex](a2a3)a1;

a2 = [tex]\lambda[/tex](a1a3)a2;

a3 = [tex]\lambda[/tex](a1a2)a3;

This gives me a1 = a2 = a3

Putting this above in constraint gives me

a1a1a1 = 1;

Therefore a1 = 1 = a2 = a3

So I am able to a1+a2+a3 = 3.

But I have not been able to prove a1+a2+a3>3. Any pointers.

Thanks

Asif

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# Homework Help: Lagrange Multipliers - 2 questions

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