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To find the critical points of the function \( f(x, y) = -8 + 5x + 5x^2 + 4y + 7y^2 \), we need to follow these steps:

### Step 1: Find the partial derivatives
We first calculate the partial derivatives of the function with respect to \(x\) and \(y\).

\[
\frac{\partial f}{\partial x} = 5 + 10x
\]
\[
\frac{\partial f}{\partial y} = 4 + 14y
\]

### Step 2: Set the partial derivatives equal to zero
To find the critical points, we set the partial derivatives equal to zero and solve for \(x\) and \(y\).

\[
5 + 10x = 0 \quad \Rightarrow \quad x = -\frac{1}{2}
\]
\[
4 + 14y = 0 \quad \Rightarrow \quad y = -\frac{2}{7}
\]

### Step 3: Critical point
Thus, the critical point is \( \left( -\frac{1}{2}, -\frac{2}{7} \right) \).

### Step 4: Classify the critical point
To classify the critical point, we use the second derivative test. We need the second-order partial derivatives:
\[
\frac{\partial^2 f}{\partial x^2} = 10, \quad \frac{\partial^2 f}{\partial y^2} = 14, \quad \frac{\partial^2 f}{\partial x \partial y} = 0
\]
The discriminant \(D\) is calculated as follows:
\[
D = \frac{\partial^2 f}{\partial x^2} \cdot \frac{\partial^2 f}{\partial y^2} - \left( \frac{\partial^2 f}{\partial x \partial y} \right)^2 = (10)(14) - 0^2 = 140
\]

Since \(D > 0\) and \( \frac{\partial^2 f}{\partial x^2} > 0 \), the critical point is a **local minimum**.

### Conclusion:
- The critical point is \( \left( -\frac{1}{2}, -\frac{2}{7} \right) \).
- It is a **local minimum**.

Would you like more details or have any questions?

Here are five related questions to consider:
1. How do you find the second partial derivatives of a function?
2. What does the discriminant tell us about a critical point?
3. How can we interpret critical points in the context of optimization problems?
4. What are the differences between local minima, maxima, and saddle points?
5. How would this method change if the function involved mixed terms like \(xy\)?

**Tip**: Always verify that the discriminant is positive before concluding a point is a minimum or maximum. If it's negative, you might have a saddle point instead!

Find the critical point of the function: f(x, y) = -8 + 5x + 5x^2 + 4y + 7y^2

Learn how to find and classify the critical point of the function f(x, y) = -8 + 5x + 5x^2 + 4y + 7y^2 using partial derivatives and the second derivative test. This problem covers key calculus concepts, including finding partial derivatives, setting them to zero, and applying the discriminant for classification. Suitable for college-level calculus students.

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