Question

find all the local maxima, local minima, and saddle points of the given function.
f(x,y)=x²+xy+y²+5x - 5y + 8
select the correct choice below and fill in any answer boxes within your choice.
a. there are local maxima located at
(simplify your answers. type ordered pairs. use a comma to separate answers as needed.)
b. there are no local maxima.
select the correct choice below and fill in any answer boxes within your choice.
a. there are local minima located at
(simplify your answers. type ordered pairs. use a comma to separate answers as needed.)
b. there are no local minima.
select the correct choice below and fill in any answer boxes within your choice.
a. there are saddle points located at
(simplify your answers. type ordered pairs. use a comma to separate answers as needed.)
b. there are no saddle points.

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.