https://file.aiquickdraw.com/mathbot/file/1729568128313av8y4mef.jpg

and I know the answer is not -34 for min or 49,50 for max

To solve the problem of finding the absolute extrema of the function \( f(x, y) = -5x^2 - 3xy + 4y^2 + 5x - 3y - 4 \) on the domain \(-2 \leq x \leq 2\) and \(-3 \leq y \leq 3\), we need to follow a systematic process that involves:

### Step 1: Finding the Critical Points
1. **First-order partial derivatives**:
   \[
   f_x = \frac{\partial}{\partial x}\left(-5x^2 - 3xy + 4y^2 + 5x - 3y - 4\right) = -10x - 3y + 5
   \]
   \[
   f_y = \frac{\partial}{\partial y}\left(-5x^2 - 3xy + 4y^2 + 5x - 3y - 4\right) = -3x + 8y - 3
   \]

2. **Set both partial derivatives to zero**:
   \[
   -10x - 3y + 5 = 0 \quad \text{(Equation 1)}
   \]
   \[
   -3x + 8y - 3 = 0 \quad \text{(Equation 2)}
   \]

   Solving this system of linear equations will yield the critical points.

### Step 2: Solving the System of Equations
From **Equation 1**:
\[
-10x - 3y + 5 = 0 \quad \Rightarrow \quad y = \frac{-10x + 5}{3}
\]

Substitute this into **Equation 2**:
\[
-3x + 8\left(\frac{-10x + 5}{3}\right) - 3 = 0
\]
Simplifying:
\[
-3x + \frac{-80x + 40}{3} - 3 = 0
\]
Multiply the whole equation by 3 to eliminate the fraction:
\[
-9x - 80x + 40 - 9 = 0 \quad \Rightarrow \quad -89x + 31 = 0
\]
\[
x = \frac{31}{89}
\]

Substitute \( x = \frac{31}{89} \) into the expression for \( y \):
\[
y = \frac{-10\left(\frac{31}{89}\right) + 5}{3} = \frac{-310 + 445}{267} = \frac{135}{267} = \frac{5}{9}
\]

Thus, the critical point is approximately \( \left( \frac{31}{89}, \frac{5}{9} \right) \).

### Step 3: Evaluate the Function at Critical Points and Boundaries
1. **Evaluate at the critical point** \( \left( \frac{31}{89}, \frac{5}{9} \right) \).

2. **Evaluate at the boundaries**: The boundaries are given by the constraints \( x = -2, 2 \) and \( y = -3, 3 \), and all combinations of these values. We also need to evaluate the function along the edges of the rectangular domain (e.g., where \( x \) is fixed and \( y \) varies or vice versa).

   For instance, along \( x = 2 \), evaluate \( f(2, y) \) as \( y \) varies between \(-3\) and \(3\). Similarly, do this for other boundary values.

### Step 4: Compare Values
Once the function has been evaluated at the critical point and along the boundaries, compare the values to find the absolute maximum and minimum.

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Would you like to go through the detailed evaluations of the boundary points and critical point, or should I proceed with just the final extrema values?

Find the absolute extrema of the function f(x, y) = -5x^2 - 3xy + 4y^2 + 5x - 3y - 4 on the domain defined by -2 ≤ x ≤ 2 and -3 ≤ y ≤ 3.

In this problem, we explore finding the absolute extrema of a function of two variables using partial derivatives and boundary analysis. The function f(x, y) = -5x^2 - 3xy + 4y^2 + 5x - 3y - 4 is evaluated on a specified domain to determine the critical points and compare values along the boundaries.

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