find all the local maxima, local minima, and saddle points of the given function.\nf(x,y)=x²+xy+y²+5x - 5y +…

Explanation:

Step1: Find the first - order partial derivatives

The function is (f(x,y)=x^{2}+xy + y^{2}+5x - 5y + 8). The first - order partial derivative with respect to (x) is (f_{x}=\frac{\partial f}{\partial x}=2x + y+5). The first - order partial derivative with respect to (y) is (f_{y}=\frac{\partial f}{\partial y}=x + 2y-5).

Step2: Find the critical points

Set (f_{x}=0) and (f_{y}=0), so we have the system of equations: (\begin{cases}2x + y+5 = 0\x + 2y-5 = 0\end{cases}) From the first equation (y=-2x - 5). Substitute (y=-2x - 5) into the second equation: (x+2(-2x - 5)-5 = 0). (x-4x-10 - 5=0). (-3x-15 = 0), then (x=- 5). Substitute (x = - 5) into (y=-2x - 5), we get (y=-2\times(-5)-5=5). So the critical point is ((-5,5)).

Step3: Find the second - order partial derivatives

(f_{xx}=\frac{\partial^{2}f}{\partial x^{2}} = 2), (f_{xy}=\frac{\partial^{2}f}{\partial x\partial y}=1), (f_{yy}=\frac{\partial^{2}f}{\partial y^{2}}=2).

Step4: Use the second - derivative test

The discriminant (D=f_{xx}f_{yy}-(f_{xy})^{2}). Substitute (f_{xx} = 2), (f_{xy}=1), (f_{yy}=2) into the formula: (D=(2\times2)-1^{2}=4 - 1=3>0). And (f_{xx}=2>0).

Answer:

A. There are local minima located at ((-5,5)); B. There are no local maxima; B. There are no saddle points.