Let's find the critical points of the given function $f(x, y) = 100 + 6xy - 4x^2 - 3y^2$, and then determine whether each critical point is a local minimum or maximum.

Step 1: Find the first partial derivatives

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

• Partial derivative with respect to $x$: $f_x = \frac{\partial}{\partial x}(100 + 6xy - 4x^2 - 3y^2) = 6y - 8x$

• Partial derivative with respect to $y$: $f_y = \frac{\partial}{\partial y}(100 + 6xy - 4x^2 - 3y^2) = 6x - 6y$

Step 2: Solve the system of equations

To find the critical points, we set both partial derivatives equal to zero: $6y - 8x = 0 \quad \text{(1)}$ $6x - 6y = 0 \quad \text{(2)}$

From equation (2), we can simplify: $6x = 6y \quad \Rightarrow \quad x = y$

Substitute $x = y$ into equation (1): $6y - 8x = 0 \quad \Rightarrow \quad 6y - 8y = 0 \quad \Rightarrow \quad -2y = 0 \quad \Rightarrow \quad y = 0$

Now that we know $y = 0$, substitute it back into $x = y$: $x = 0$

Thus, the only critical point is $(0, 0)$.

Step 3: Determine whether it's a local minimum or maximum

To classify the critical point, we need to use the second derivative test. We calculate the second-order partial derivatives:

• $f_{xx} = \frac{\partial^2 f}{\partial x^2} = -8$
• $f_{yy} = \frac{\partial^2 f}{\partial y^2} = -6$
• $f_{xy} = \frac{\partial^2 f}{\partial x \partial y} = 6$

The discriminant $D$ is given by: $D = f_{xx} f_{yy} - (f_{xy})^2$ Substitute the values: $D = (-8)(-6) - (6)^2 = 48 - 36 = 12$

Since $D > 0$ and $f_{xx} < 0$, the function has a local maximum at the critical point $(0, 0)$.

Final Answer:

The critical point is $(0, 0)$, and it is a local maximum.

Would you like more details, or do you have any questions about this solution?

Here are 5 related questions:

1. How do you find the second-order partial derivatives of a function?
2. What are the conditions for a critical point to be a local minimum or maximum?
3. How does the discriminant $D$ relate to the classification of critical points?
4. What happens if the discriminant $D$ is negative?
5. How do you compute higher-order derivatives for multivariable functions?

Tip: Always check the sign of the discriminant $D$ and the second derivatives to accurately classify critical points.