Find the minimum of f(x, y) = (x - y)2 + ( √(2 - x2) - 9/y )2 in the half-infinite strip 0 < x < √2, y > 0.
Setting the partial derivatives to zero gives: y2 = 9x/√(2 - x2) and -x + y + 9/y2 √(2 - x2) - 81/y3. Using the first equation to substitute 9x/y2 for √(2 - x2), the second equation becomes (x - y)(81/y4 - 1), so x = y or y = 3. Substituting back in the first equation, we find that x = y does not give a stationary point, whereas y = 3 gives x = 1, y = 3. This gives the value 8. Finally, we check that we cannot get a minimum on the edges y = 0, x = √2, and that the value is a minimum not a maximum or saddle point by comparing the value to two nearby values.
45th Putnam 1984
© John Scholes
7 Jan 2001