Krazy G's question at Yahoo! Answers regarding extrema for a function of two variables

MarkFL

Staff member
Here is the question:

Find the (x; y) coordinates of the stationary point of: z(x,y) = 2x^2 - 2xy + y^2 + 2x + 5 and find the natu?

Find the (x; y) coordinates of the stationary point of:

z(x,y) = 2x^2 - 2xy + y^2 + 2x + 5

and find the nature of the stationary point.
I have posted a link there to this thread so the OP can view my work.

MarkFL

Staff member
Hello Krazy G,

We are given the function:

$$\displaystyle z(x,y)=2x^2-2xy+y^2+2x+5$$

First, we want to find the critical points by equating the first partials to zero:

$$\displaystyle z_x(x,y)=4x-2y+2=0$$

$$\displaystyle z_y(x,y)=-2x+2y=0$$

The second equation implies $y=x$, and substitution for $y$ into the first equation yields:

$$\displaystyle x=-1$$

and so the critical point is:

$$\displaystyle (x,y)=(-1,-1)$$

Now, to determine the nature of this critical point we may utilize the second partials test for relative extrema.

$$\displaystyle D(x,y)=z_{xx}(x,y)z_{yy}(x,y)-\left[z_{xy}(x,y) \right]^2=4\cdot2-(-2)^2=4$$

Since $$\displaystyle z_{xx}(x,y)=4>0$$ and $$\displaystyle D(x,y)=4>0$$, then we conclude that the critical value is the global minimum. Hence:

$$\displaystyle z_{\min}=z(-1,-1)=4$$