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

In summary, the task was to calculate the gradient, directional derivative, and maximum value of a function at the point (2,3,1), but the instructions were unclear.
  • #1
trelek2
88
0
given:[tex]
f(r)=x ^{2}+3y ^{2} +2z ^{2} [/tex]
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.
[tex] r ^{2} =1[/tex]
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.
 
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  • #2
trelek2 said:
given:[tex]
f(r)=x ^{2}+3y ^{2} +2z ^{2} [/tex]
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.
[tex] r ^{2} =1[/tex]
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 [itex]r= x^2+ y^2+ z^2= 1[/itex], although I would be inclined to use "[itex]\rho[/itex]", or [itex]r= x^2+ y^2= 1[/itex].
 
  • #3
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"
 
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  • #4
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?
 
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  • #5
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.
 
  • #6
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?
 

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

What is the meaning of "Max Value f(r): Find Solution w/ Condition r^2=1"?

This phrase refers to a mathematical problem where a function, represented by f(r), needs to be maximized while also satisfying the condition that r^2=1. This means that the variable r must have a value of either 1 or -1.

What is the importance of finding the maximum value of a function?

Finding the maximum value of a function is important in many fields of science, as it helps determine the most optimal solution to a problem. In physics, for example, finding the maximum value of a force can help determine the most efficient way to move an object. In economics, finding the maximum value of profit can help determine the best business strategy.

How can I find the maximum value of a function with a given condition?

There are various methods for finding the maximum value of a function with a given condition. One approach is to use calculus and take the derivative of the function, set it equal to zero, and solve for the variable. Another approach is to graph the function and visually determine the maximum point while also considering the given condition.

Are there any real-world applications of this mathematical problem?

Yes, there are many real-world applications of this problem. For example, in engineering, finding the maximum value of a force while also satisfying a certain condition can help determine the most efficient design for a structure. In statistics, this problem can be used to find the most accurate prediction model while also considering a given constraint.

What other conditions can be added to this problem?

This problem can be modified by adding different conditions to the function, such as r^2=2 or r^2=3. Additionally, multiple conditions can be added, making the problem more complex. The specific conditions added will depend on the context and purpose of the problem.

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