Find absolute maximum and minimum values

Question

Find absolute maximum and minimum values of f(x, y) = x^2 + 14xy + y^2, defined on the disc D = {(x, y)|x^2 + y^2 ≤ 7}.

maximum value : ?

minimum value : ?

Answer 1

Procedure is to find critical points for del f = (0,0) which in this case is (2x+14y, 2y+14x) solve the two equations simultaneously to obtain (0,0) crit. pt. This is within the bounds...Calculate the Hessian:

f_xxf_yy - f_xy² = 2*2 - 14² <0 saddle point. which means that the min/max will be on the boundary.

To work out maxima on the boundary, you can express f(x,y) as f(x) = 14x*sqrt(7-x²) + 7 by plugging in (y = sqrt(7-x²)

Now you can take the derivative with respect to x. Don't for get that the solutions will be +/- for x and for y. These will be the global maxes and mins.