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.

Explanation:

Step1: Find first - order partial derivatives

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

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

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:
\(3\times\frac{6 - 3y}{4}+8y+7 = 0\)
\(\frac{18-9y}{4}+8y+7 = 0\)
Multiply through by 4: \(18-9y + 32y+28 = 0\)
\(23y=-46\), so \(y = - 2\)
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

\(t_{xx}=\frac{\partial^{2}t}{\partial x^{2}}=4\), \(t_{xy}=\frac{\partial^{2}t}{\partial x\partial y}=3\), \(t_{yy}=\frac{\partial^{2}t}{\partial y^{2}}=8\)

Step4: Use the second - derivative test

The discriminant \(D=t_{xx}t_{yy}-(t_{xy})^{2}\)
\(D=(4\times8)-3^{2}=32 - 9=23>0\) and \(t_{xx}=4>0\)

Answer:

B. There are no local maxima.