Econ 407: Finding the critical values of functions of two variables.

Recall that the first and second derivatives are routinely used to find the critical points of a function. The procedure is similar for functions with two or more variables. The only diffrence is that we use partial derivatives in order to investigate the rate of change of the function with respect to one variable at a time while holding other variables constant. We will use the following function to illustrate these points. Please note that Zx stands for the first partial derivative, Zxx stands for the second partial derivative, and Zxy stanads for the cross partial derivatives. The conditions for establishing critical values are noted below.

> restart;

> Z:=5*x^3-3*x^2*y^2+7*y^5;

> Zx:=diff(Z,x);

> Zy:=diff(Z,y);

> Zxy:=diff(Zx,y);

> Zyx:=diff(Zy,x);

> z:=2*y^3-x^3+147*x-54*y+12;

> plot3d(z,x=-10..10,y=-10..10);

> zx:=diff(z,x);

> zxx:=diff(zx,x);

> zy:=diff(z,y);

> zyy:=diff(zy,y);

> solve({zx,zy},{x,y});

> eval(zxx,{x=7,y=3});

> eval(zyy,{x=7,y=3});

> zyx:=diff(zy,x);

> eval(zxx*zyy,{x=7,y=3});

> 36*(-42);

x=7, y=3 can not be a maximum, or a minimum because the product of the second deritivates is negative in (-1512), which is less than zero. Therefore the test is inconclusive. In this case we have a saddle point. At a saddle point, the function will show a minimum when viewed from one axis and a maximum when viewed from another. Let us evaluate the function for the other roots.

> eval(zxx,{x=7,y=-3});

> eval(zyy,{x=7,y=-3});

> eval(zxx*zyy,{x=7,y=-3});

The product of the second deritavites is positive and equal to 1512, which is greater than the squared cross-deritavite, which is 0. Therefore, we have a relative minimum.

> eval(zxx,{x=-7,y=3});

> eval(zyy,{x=-7,y=3});

>

> eval(zxx*zyy,{x=-7,y=3});

Conditions for relative maximum:

1. Zx and Zy=0

2. Zxx and Zyy <0

3. (Zxx)*(Zyy)>(Zxy)^2

Conditions for relative minimum:

1. Zx and Zy=0

2. Zxx and Zyy >0

3. (Zxx)*(Zyy)>(Zxy)^2

Importnat points : if (Zxx)*(Zyy)<(Zxy)^2 and Zxx and Zyy have the same signs, the function is at an inflection point. When Zxx and Zyy have different signs, the function is at a saddle point. Note that when Zxx and Zyy have different signs, their products cannot be greater than (Zxy)^2, so the function has to be at a saddle point.