You say you used Langrange multipliers but you don't show how you did that. All I can say is that, using Lagrange multipliers, I get (0, 5, 0).
"Find the point on the paraboloid :
z=(x^2)/25 + (y^2)/4
That is closest to the point (0,5,0)"
...my solutions manual solves this using symetry. they say that by symetry we see that x=0. And then the problem turns into a two dimensional problem. and optimize the distance squared... they got (0,2.66,1.76) as the closest point.
...NOW, i did this in 3D using the paraboloid as my constraint, and did it by the method of lagrange multipliers. but i get a totally diff answer were some of my points are negative.
Can someone shed some light please. Thanks in advance
I really hope i did some dumb mistake bc this problem is frustating me.
Ok, my function to be optimized is the square of the distance formula, between the (0,5,0) and a point x,y,z on the paraboloid: d^2= F = x^2 + (y-5)^2 + z^2 and my constraint will be the parabloid function set equal to zero: (x^2)/25 + (y^2)/4 - z = 0. and then i did the partial integrations, setting them equal to the partials of the constraint times lambda. Is there something wrong in my analysis?
Your are right. I don't know where my head was! (0, 5, 0) was the point we are measuring the distance to, not the answer.
The problem is to minimize subject to the constraint . or , , and .
If you divide the first equation by the last, so x= 0 or . If you divide the first equation by the second, so so xy= 20.