Exam : find the minimum value of this function and constrant

In summary, RGV explains that the problem can be solved by substituting y = c-x into the equation for f, and minimizing the resulting univariate function. F is strictly convex, so any stationary point is a global minimum. However, solving for x is a bit tricky, and Maple 11 was used to find the solution.
  • #1
~Scott~
5
0

Homework Statement


Given
[tex] x+y = 6\sqrt2[/tex]
[tex] a + b = 6\sqrt 2 [/tex]
Find minimum value of [tex]\sqrt{x^2 + a^2} + \sqrt{y^2+b^2}[/tex]

2. The attempt at a solution
I had plotted the [tex] f(x,y) [/tex] with constant [tex] a ,b[/tex]
And that's all. I don't realize how to do this.

Any comment will be appreciated
 
Last edited:
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  • #2
I don't know where on Earth you found this question, but one can find the minimum value. The key step is to notice that you can simplify the total derivative considerably using the relation between x and y. That, together with...

But I've probably said enough. I figure this is a take-home exam or something.
 
Last edited:
  • #3
If this is a take home exam, we really shouldn't help you.
 
  • #4
It's a Lagrange multiplier problem with two constraints. Try that technique.
 
  • #5
Let me introduce myself. Sorry to take pity on me.
I graduated from Physics and now studying in computer science.

Of course, I studied Lagrange multiplier in classical mechanics.

I always spend my free time to enjoy mathematics and computing.
I found this problem from math.or.th age not over 15.
And I never expect that I have to use Lagrange multiplier.

You can check my post about GPU/CUDA topics that is my first post.
And now it's finish without any reply from this forum.

And second post of mine is about gaussian gun. Again no one reply.

The third one is this about lagrange multiplier.

Guess what.
Your people totally look down on me just because you known how to solve it
What about two questions of mine ?

I know your situation that many lazy students exploit this forum for their benefit.
It's ok, but please count me out.

If anyone would like me to identify myself as a master degree in computer science. Just sent me your PM and I will show it to you.

Thanks everyone who show me the topics of Lagrange multiplier.
 
  • #6
~Scott~ said:
Let me introduce myself. Sorry to take pity on me.
I graduated from Physics and now studying in computer science.

Of course, I studied Lagrange multiplier in classical mechanics.

I always spend my free time to enjoy mathematics and computing.
I found this problem from math.or.th age not over 15.
And I never expect that I have to use Lagrange multiplier.

You can check my post about GPU/CUDA topics that is my first post.
And now it's finish without any reply from this forum.

And second post of mine is about gaussian gun. Again no one reply.

The third one is this about lagrange multiplier.

Guess what.
Your people totally look down on me just because you known how to solve it
What about two questions of mine ?

I know your situation that many lazy students exploit this forum for their benefit.
It's ok, but please count me out.

If anyone would like me to identify myself as a master degree in computer science. Just sent me your PM and I will show it to you.

Thanks everyone who show me the topics of Lagrange multiplier.

There is another way: substitute y = c-x into f(x,y), where c = 6*sqrt(2). The result F(x) = f(x,c-x) is a univariate function, and can be minimized by setting its derivative to zero; in fact, we know the function F is strictly convex, so any stationary point is a global minimum.

Finding the derivative is not too hard, but solving the equation F'(x) = 0 for x is a bit nasty. (Here, I am assuming that a and b are not variables, but are just some constants that satisfy a+b=c.) It turns out that the solution is just x = a (when the relationship between a and b is exploited). I took the easy way out, and let Maple 11 solve the equations for me; doing the problem by hand would be difficult.

RGV
 
  • #7
Thank you RGV.
I will check it.
 

1. What is the purpose of finding the minimum value of a function?

The minimum value of a function is the lowest point on its graph, which can provide important information about the behavior and characteristics of the function. It can also be used to optimize the function for specific applications.

2. How is the minimum value of a function calculated?

The minimum value of a function can be found by taking the derivative of the function and setting it equal to zero. This will give the x-coordinate of the minimum point, which can then be substituted back into the original function to find the minimum value.

3. Is there a specific method for finding the minimum value of a function?

There are several methods for finding the minimum value of a function, including using calculus, graphical analysis, and algebraic techniques such as completing the square. The most appropriate method depends on the complexity and type of function.

4. What is a constraint and how does it affect finding the minimum value of a function?

A constraint is a condition or limitation that must be satisfied in order to find the minimum value of a function. It can be represented as an inequality or equation that restricts the possible values of the independent variable. Constraints can make the process of finding the minimum value more complex, as they must be taken into account during the calculation.

5. Can a function have more than one minimum value?

Yes, a function can have multiple minimum values if it has multiple local minimum points. These points are typically separated by maximum points or points of inflection. It is important to specify the domain and range of the function in order to determine if there are multiple minimum values.

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