Find the maximum and minimum values of the function f (x, y) = x ^ y (4-x-y) in the closed region D bounded by the straight line x + y = 6, y = 0, x = 0

Find the maximum and minimum values of the function f (x, y) = x ^ y (4-x-y) in the closed region D bounded by the straight line x + y = 6, y = 0, x = 0

The method is to divide the region into two parts: in the interior, we use the sufficient conditions of extremum, that is, to find the stationary point, and then we use the three second-order partial derivatives to verify the sufficient conditions of extremum; in the boundary, we use the conditional extremum, and we can substitute x = 0, y = 0, x = 6-y into the extremum of univariate function

Given the circle C (x-1) ^ 2 + (Y-1) ^ 2 = 4, cross the point m (0,1) to make a straight line L and cross the circle C at two points AB to find the maximum area of △ ABC

If the center coordinate (1,1) passes through M (0,1), then it passes through the center, that is, AB is the diameter of the circle! The maximum area is that point C is directly below,
C (1, - 1)
S=4 x 2 x 1/2 =4

It is known that the circle C: x2-8x + y2-9 = 0, passing through the point m (1,3) makes a straight line intersection circle C at two points a and B, and the maximum area of △ ABC is___ ．

Let the linear equation passing through the point m (1,3) be l: Y-3 = K (x-1), the center of the circle C (4,0) is obtained from x2-8x + y2-9 = 0, and the radius r = 5. Let the distance between the center of the circle C (4,0) and the line L be D, and the projection of point C on l be m, then d = 3|1 + k|1 + K2; in the right angle △ CMA, (|ab|2) 2 = R2-D2 = 25-9 (1 + k) 21 + K2 = 16 -