Question

find all points where the function has any relative extrema. identify any saddle points.
f(x,y)=2x² + 3xy + 4y² - 5x + 2y
find all points where the function has relative maxima. select the correct choice below and fill in any answer boxes within your choice.
a. the point(s) is(are) ▢.
(simplify your answer. type an ordered pair. use a comma to separate answers as needed.)
b. there are no relative maxima.
find all points where the function has relative minima. select the correct choice below and fill in any answer boxes within your choice.
a. the point(s) is(are) ▢.
(simplify your answer. type an ordered pair. use a comma to separate answers as needed.)
b. there are no relative minima.
identify any saddle points. select the correct choice below and fill in any answer boxes within your choice.
a. there is(are) saddle point(s) located at ▢.
(simplify your answer. type an ordered pair. use a comma to separate answers as needed.)
b. there are no saddle points.

Explanation:

Step1: Find first - order partial derivatives

The function is \(f(x,y)=2x^{2}+3xy + 4y^{2}-5x + 2y\).
The first - order partial derivative with respect to \(x\) is \(f_{x}=\frac{\partial f}{\partial x}=4x + 3y-5\).
The first - order partial derivative with respect to \(y\) is \(f_{y}=\frac{\partial f}{\partial y}=3x + 8y+2\).

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

We have the system of equations \(

$$\begin{cases}4x + 3y-5 = 0\\3x + 8y+2 = 0\end{cases}$$

\).
From the first equation \(4x+3y=5\), we get \(x=\frac{5 - 3y}{4}\).
Substitute \(x=\frac{5 - 3y}{4}\) into the second equation \(3\times\frac{5 - 3y}{4}+8y+2 = 0\).
Multiply through by \(4\) to get \(3(5 - 3y)+32y + 8 = 0\).
Expand: \(15-9y+32y + 8 = 0\).
Combine like terms: \(23y=-23\), so \(y=-1\).
Substitute \(y = - 1\) into \(x=\frac{5 - 3y}{4}\), then \(x=\frac{5-3\times(-1)}{4}=\frac{5 + 3}{4}=2\).

Step3: Find second - order partial derivatives

\(f_{xx}=\frac{\partial^{2}f}{\partial x^{2}} = 4\), \(f_{xy}=\frac{\partial^{2}f}{\partial x\partial y}=3\), \(f_{yy}=\frac{\partial^{2}f}{\partial y^{2}}=8\).

Step4: Use the second - derivative test

The discriminant \(D=f_{xx}f_{yy}-(f_{xy})^{2}\).
Substitute \(f_{xx}=4\), \(f_{xy}=3\), \(f_{yy}=8\) into \(D\): \(D=(4\times8)-3^{2}=32 - 9=23>0\).
And \(f_{xx}=4>0\).

Answer:

For relative minima: A. The point(s) is(are) \((2,-1)\).
For local relative: B. There are no local relative.
For saddle points: B. There are no saddle points.