Find the extreme values of f on the region described by the inequality. f(x, y) = x2 + y2 + 6x − 6y, x2 + y2 ≤ 49

The function has critical points where the first partial derivatives simultaneously vanish:




The point (3, 3) lies within the disk . The Hessian for is



which is positive definite for all . By the second partial derivative test, this indicates the critical point (3, 3) is the site of a minimum, and we have .

Meanwhile, when we consider the boundary, we can set and , so that can be expressed as a function of one variable :



Then the critical points of occur at



where is any integer. We restrict to omit redundancies, leaving us with just and . We then have





which indicates that a minimum occurs at , and a maximum at . At these points we have values of and .

So the absolute minimum of over the disk is -18, and the absolute maximum of over the disk is .