find all points where the given function has any relative extrema. identify any saddle points.\n$f(x,y)=-5x^{…

Explanation:

Step1: Find first - order partial derivatives

The first - order partial derivatives of (z = f(x,y)=-5x^{2}-6xy - 5y^{2}-44x-20y + 8) are: (f_{x}=\frac{\partial f}{\partial x}=-10x-6y - 44) (f_{y}=\frac{\partial f}{\partial y}=-6x-10y - 20)

Step2: Solve the system of equations (f_{x}=0) and (f_{y}=0)

We have the system of equations: (\begin{cases}-10x-6y-44 = 0\-6x - 10y-20=0\end{cases}) From the first equation ( - 10x-6y=44), we get (y=\frac{-10x - 44}{6}) Substitute (y=\frac{-10x - 44}{6}) into the second equation: (-6x-10\times\frac{-10x - 44}{6}-20 = 0) Multiply through by (6) to clear the fraction: (-36x+100x + 440-120 = 0) (64x=-320), so (x=- 5) Substitute (x = - 5) into (y=\frac{-10x - 44}{6}), (y=\frac{-10\times(-5)-44}{6}=\frac{50 - 44}{6}=1)

Step3: Find second - order partial derivatives

(f_{xx}=\frac{\partial^{2}f}{\partial x^{2}}=-10), (f_{xy}=\frac{\partial^{2}f}{\partial x\partial y}=-6), (f_{yy}=\frac{\partial^{2}f}{\partial y^{2}}=-10)

Step4: Use the second - derivative test

The discriminant (D=f_{xx}f_{yy}-(f_{xy})^{2}) (D=(-10)\times(-10)-(-6)^{2}=100 - 36 = 64) Since (D>0) and (f_{xx}=-10<0)

Answer:

A. The relative maximum/maxima is/are located at ((-5,1))