Solve Optimization Problem: Find Min & Max of f(x,y)

[Yankel]

Hello all

I am trying to find minimum and maximum of the following function:

\[f(x,y)=4x^{2}-y^{2}-xy-2x+6y\]

under the constraints:

\[y=4-2x\]

\[x\geq 0\]

\[y\geq -2\]I tried solving this problem using the method of the method of bounded and closed domain, understanding that the constraints creates a triangle. I checked every line in the triangle, the edge points and the local min and max for each line (if there were any).

I got that the minimum value was f(0,-2)=-16 and the maximum was f(3,-2)=20
(should I have used Lagrange multipliers ?)

The problem is:

1. I entered this problem to MAPLE, and got max like mine, but min at f(0.5,3)=7.5. I found this point, but it isn't the absolute minimum.

2. In the answers sheet for this problem there are 4 possible answers for the sum of the min+max: 30.5, -7, 0. 16.
Non of them are according to my solution or MAPLE's.

Can you please assist me with solving this problem ?

Thank you !

 

[MarkFL]

Using Lagrange multipliers, I find:

\(\displaystyle f_{\min}=\frac{15}{2}\)

\(\displaystyle f_{\max}=20\)

Post your work using Lagrange, and I will be glad to look it over. :D

 

[Yankel]

I didn't use Lagrange, I looked at it as an optimization problem in a closed region, using a triangle.

I notice that the point (0,-2) which is my smallest, is not even in the region the constraint creates. Therefore, if I ignore it, I get the same results you got using Lagrange. This means that the published answers are wrong (if you, me and MAPLE say the same thing...).

 

[Fallen Angel]

Hi,

It's even easier if you simply use the equality constrain to go through a single valued function and derive the function.

I agree with your answers, so maybe the provided ones are wrong.