Question
Suppose f(x,y) = xy(1-1x-4y)
f(x,y) has 4 critical points. List them in increasing lexographic order. By that we mean that (x, y) comes before (z, w) if x<z or if x=z and y<w . Also, describe the type of critical point by typing MA if it is a local maximum, MI if it is a local minimim, and S if it is a saddle point.
First point (__,__ ) of type ___
Second point (__,__ ) of type ____
Third point (__,__ ) of type ___
Fourth point (__,___ ) of type ___
Explanation / Answer
uppose f(x, y) = xy(1 - 1 x - 7 y). f(x, y) has 4 critical points. List them in increasing lexographic order. By that we mean that (x, y) comes before (z, w) if x (0, 1/7) x = 1-14y, y = 0 -----> (1, 0) x = 1-14y, 7y = 1-2x -----> (1/3, 1/21) ==================== fxx = ?/?x (fx) = -2y fyy = ?/?y (fy) = -14x fxy = ?/?y (fx) = 1 - 2x - 14y M(x,y) = fxx * fyy - (fxy)² M(x,y) = 28xy - (1-2x-14y)² For critical points (a,b): If M(a,b) > 0 and fxx > 0, then (a,b) is local minimum If M(a,b) > 0 and fxx < 0, then (a,b) is local maximum If M(a,b) < 0, then (a,b) is saddle point If M(a,b) = 0, then result is inconclusive M(0,0) = 0 - (1)² = -1 < 0 -----> saddle point M(0, 1/7) = 0 - (1-0-2)² = -1 < 0 -----> saddle point M(1,0) = 0 - (1-2-0)² = -1 < 0 -----> saddle point M(1/3,1/21) = 28(1/3)(1/21) - (1-2/3-2/3)² = 1/3 > 0 . . . and fxx(1/3,1/21) = -2(1/21) = -2/21 < 0 -----> local maximum ==================== Solutions: (0, 0) of type S (0, 1/7) of type S (1/3, 1/21) of type MA (1, 0) of type S