The problem requires finding the critical points of the given function $f(x, y) = 2x^2 + 3y^2 + 9y + 2x + 5xy$, and determining whether the critical points correspond to a local minimum, maximum, or saddle point. Let's proceed step by step:

Step 1: Compute Partial Derivatives

To find the critical points, we need the partial derivatives of $f(x, y)$ with respect to $x$ and $y$:

$\frac{\partial f}{\partial x} = 4x + 5y + 2$ $\frac{\partial f}{\partial y} = 6y + 5x + 9$

Step 2: Solve for Critical Points

Set the partial derivatives to zero:

1. $\frac{\partial f}{\partial x} = 0$:
   $4x + 5y + 2 = 0$

2. $\frac{\partial f}{\partial y} = 0$:
   $6y + 5x + 9 = 0$

We now solve this system of equations for $x$ and $y$. Substituting $x$ or $y$ from one equation into the other will provide the critical points.

Step 3: Analyze Critical Points

Use the second derivative test to determine the nature of the critical points. First, compute the second partial derivatives:

• $f_{xx} = \frac{\partial^2 f}{\partial x^2} = 4$
• $f_{yy} = \frac{\partial^2 f}{\partial y^2} = 6$
• $f_{xy} = f_{yx} = \frac{\partial^2 f}{\partial x \partial y} = 5$

The Hessian determinant is given by: $H = f_{xx} f_{yy} - (f_{xy})^2 = (4)(6) - (5)^2 = 24 - 25 = -1$

Since $H < 0$, the critical point (once solved) corresponds to a saddle point.

Let me calculate the critical points explicitly. sympy import symbols, Eq, solve

Define variables

x, y = symbols('x y')

Partial derivatives set to zero

partial_x = 4x + 5y + 2 partial_y = 6y + 5x + 9

Solve the system of equations

critical_points = solve([Eq(partial_x, 0), Eq(partial_y, 0)], (x, y)) critical_pointsThe critical point is $(-33, 26)$.

Step 4: Nature of the Critical Point

As previously calculated, the Hessian determinant ($H = -1$) is negative, which means the critical point $(-33, 26)$ is a saddle point.

Would you like a detailed explanation of any specific step? Here are some related questions:

1. How do you compute second partial derivatives for multivariable functions?
2. What is the Hessian determinant, and how does it determine the nature of a critical point?
3. Why is a saddle point neither a maximum nor a minimum?
4. Can this method be extended to higher dimensions (e.g., functions of three variables)?
5. How would graphing this function confirm the nature of the critical point?

Tip: Always verify the determinant of the Hessian to confirm your interpretation of critical points, especially when analyzing complex functions.