- #1
asif zaidi
- 56
- 0
Hello:
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
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