Question

The function \[ f(x, y)=x y(1-5 x-9 y) \] has 4 critical points. List them and select the type of critical point. Points should be entered as ordered pairs and listed in increasing lexicogra.

Answer 1

These solutions are found in increasing lexicographic order as [ (0,0), \quad (-\frac{1}{9},0), \quad (0,-\frac{1}{5}), \quad \left(\frac{9}{40}, \frac{3}{80}\right), ] and the critical points are degenerate, degenerate, saddle, and local maximum, respectively.

Step-by-step explanation:

To determine the nature of each critical point, we can compute the Hessian of f, [ (f_{xx}, f_{xy}) = y(-45x+9), \qquad (f_{xy}, f_{yy}) = x(-9y+5). ] Thus, detF=−45xy<0 at each critical point except (0,0). (Recall that a critical point where detF is nonzero is either a local minimum or a local maximum, and a critical point where detF=0 is either a saddle point or a degenerate critical point.)

Note that (0,0) is the origin of the plane, so there is no f(x,y) value near there that makes both of its partial derivatives 0. So (0,0) is a degenerate critical point. If a coordinate is non-zero at a critical point, then multiplying the function by that coordinate and reversing that coordinate does not change the function near that point. For example, we note that f(x,y)=9xf(x,y/9), so when y=0, one of the factors of f(x,y) is at most 0; but when y=0 none of the factors of f(x,y) is at most 0. So (−1/9,0) is also a degenerate critical point, by similar logic.