# Local maxima and minima

1. Jun 13, 2014

### clairaut

A rectangle with length L and width W is cut into four smaller rectangles by two lines parallel to the sides. Find the maximum and minimum values of the sum of the squares of the areas of the smaller rectangles.

Unless I did incorrectly, the algebra is very very long...

HELP

2. Jun 13, 2014

### clairaut

[A(x,y)]= 4x^2y^2 + 2L^2y^2 - 4Lxy^2 - 4wx^2y + 2w^2x^2 + 4wLxy - 2w^2Lx - 2wL^2y + w^2L^2

Constraint C= WL = area1 = constant

3. Jun 13, 2014

### pasmith

Firstly, this belongs in the homework forum.
Secondly, this problem does not involve lagrange multipliers.
Thirdly, it's best to factorize $A(x,y)$ thusly: $$A(x,y) = x^2 y^2 + x^2(W-y)^2 + (L - x)^2y^2 + (L-x)^2(W-y)^2 = (x^2 + (L-x)^2)(y^2 + (W-y)^2).$$ Both factors are non-negative in the domain we're interested in, so the maximum of the product is the product of the maxima, and the minimum of the product is the product of the minima.

4. Jun 17, 2014

### clairaut

Thanks!

When i do the first and second derivative test, I can find the local minimums.

However, I can only deduce the local maximum without a formal derivative test. Is there a way to mathematically prove the local maxima?