Question

find all the local maxima, local minima, and saddle points of the given function.
f(x,y)=2x² + 3xy + 4y² - 6x + 7y
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.

Explanation:

Step1: Find first - order partial derivatives

The first - order partial derivatives of \(f(x,y)=2x^{2}+3xy + 4y^{2}-6x + 7y\) are:
\(f_{x}=\frac{\partial f}{\partial x}=4x + 3y-6\)
\(f_{y}=\frac{\partial f}{\partial y}=3x + 8y+7\)

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

We have the system of equations:
\(

$$\begin{cases}4x + 3y-6 = 0\\3x + 8y+7 = 0\end{cases}$$

\)
From the first equation \(4x+3y=6\), we get \(x=\frac{6 - 3y}{4}\)
Substitute \(x=\frac{6 - 3y}{4}\) into the second equation:

$$
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$$

Substitute \(y = - 2\) into \(x=\frac{6 - 3y}{4}\), we get \(x=\frac{6-3\times(-2)}{4}=\frac{6 + 6}{4}=3\)

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: Calculate the discriminant \(D=f_{xx}f_{yy}-(f_{xy})^{2}\)

\(D=(4\times8)-3^{2}=32 - 9=23>0\) and \(f_{xx}=4>0\)

Answer:

A. There are local minima located at \((3,-2)\). B. There are no local maxima.