Max Value f(r): Find Solution w/ Condition r^2=1

[trelek2]

given:<br /> f(r)=x ^{2}+3y ^{2} +2z ^{2}
The task was to calculate at the point (2,3,1): the grad of f, tangent plane, directional derivative in the direction (2,-1,0) but also to find the maximum value of f subject to the condition that.
r ^{2} =1
I've done all except the last part, I have no idea what I am supposed to do here, and I don't really understand what they want.
Please explain.

 

[HallsofIvy]

given:<br /> f(r)=x ^{2}+3y ^{2} +2z ^{2}
The task was to calculate at the point (2,3,1): the grad of f, tangent plane, directional derivative in the direction (2,-1,0) but also to find the maximum value of f subject to the condition that.
r ^{2} =1
I've done all except the last part, I have no idea what I am supposed to do here, and I don't really understand what they want.
Please explain.

Well, I don't either because there is not "r" given. If I had to guess it would be either r= x^2+ y^2+ z^2= 1, although I would be inclined to use "\rho", or r= x^2+ y^2= 1.

 

[yyat]

Why do you write "f(r)" when f is a function of x,y,z and can not be written as a function of the radius?
To find the maximum you should probably use the Langrange multiplier method (find points where gradient is normal to the set on which f should be optimized).
Link: http://en.wikipedia.org/wiki/Lagrange_multipliers"

 

[trelek2]

It is what i have in the exercise...
I also have problems distinguishing r and the x+y+z stuff. How would you treat it?
So supposing r^2=1=x^2+y^2+z^2, should I then take the gradient of f(r) at the given point to find the value of the langrange multiplayer?

 

[yyat]

Apply the Langrange multiplier method (as in the wikipedia article or maybe in your notes/textbook) to the function f(x,y,z)=x^2+3y^2+2z^2 and the constraint g(x,y,z)=x^2+y^2+z^2=1. You have x,y,z instead of just x,y as in the wikipedia article, but you should be able to adapt the formulas easily.

 

[trelek2]

I'm still really confused how to do this. Since I get the gradient of f and the constraint function g in terms of (2x+2xlambda, 6y+2ylambda,4z+2zlambda) It seems to imply that lambda has to be 3 different values at the same time as the variables get reduced. To keep the variables I can take as stated in the exercise at the point f(2,3,1) but does that make any sense?