Lagrange Multiplier Question

Find the rectangle in the ellipse  x²+4y² =1 with the smallest perimeter. 

I know that if L=f(x,y) + λ g(x,y) ,

then λ g(x,y) = λ (x²+4y² -1)

However the solution states that f(x,y)=4x+4y. 

Why doesn’t 2x+2y denote the perimeter of the rectangle? Or is there a mistake in the solution?

Thanks

Student X

4x+4y should denote the perimeter of the rectangle.Think for a moment, the ellipse is one which is clearly centered at the origin.

Hence,the length and breath of the rectangle would have to span both the -ve/+ve x and y axes in a symmetrical fashion to sit snugly within the ellipse.

The horizontal and vertical sides of the rectangle would be represented by 2x and 2y respectively.

Hence, the perimeter would be 2*(2x+2y)= 4x+4y.

Hope this clarifies. Peace.

Best Regards,

Mr Koh