Question

Let y=2-x^2. Choose a point (x,y) in the first quadrant that is on this curve. Then form a rectangle in the first quadrant using the points (x,y) and (0,0) as the ends of a diagonal for the rectangle. Find the values of x and y which make the area of the rectangle the largest.

Answer 1

area of rectangle is xy
A=x(2-x^2)
=2x-x^3
diff A w.r.t x to get extremum as x=\[\sqrt{2/3}\]
==>y=4/3

Answer 2

maximize f(x,y) = xy for constraint y = 2 - x^2
might be this http://www.wolframalpha.com/input/?i=maximize+x%282-x^2%29

http://www.wolframalpha.com/input/?i=2+-+2%2F3 says it lies in first quadrant