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This problem asks to find the stationary points of the function \( f(x, y) = 9x^2 - 6xy + y^2 \) subject to the constraint \( g(x, y) = x^2 + y^2 - 25 = 0 \). This is a classical optimization problem that can be solved using **Lagrange multipliers**.

### Step 1: Form the Lagrange function
The method of Lagrange multipliers requires us to define a Lagrange function \( \mathcal{L} \) as follows:

\[
\mathcal{L}(x, y, \lambda) = f(x, y) - \lambda g(x, y)
\]

Substitute the given functions:

\[
\mathcal{L}(x, y, \lambda) = (9x^2 - 6xy + y^2) - \lambda (x^2 + y^2 - 25)
\]

### Step 2: Compute the partial derivatives
Now, take the partial derivatives of \( \mathcal{L} \) with respect to \( x \), \( y \), and \( \lambda \):

1. \( \frac{\partial \mathcal{L}}{\partial x} = 18x - 6y - 2\lambda x = 0 \)
2. \( \frac{\partial \mathcal{L}}{\partial y} = -6x + 2y - 2\lambda y = 0 \)
3. \( \frac{\partial \mathcal{L}}{\partial \lambda} = -(x^2 + y^2 - 25) = 0 \)

From the third equation, we recover the constraint:

\[
x^2 + y^2 = 25
\]

### Step 3: Solve the system of equations
We now have the following system of equations:

1. \( 18x - 6y - 2\lambda x = 0 \)
2. \( -6x + 2y - 2\lambda y = 0 \)
3. \( x^2 + y^2 = 25 \)

We will solve these equations simultaneously to find the values of \( x \), \( y \), and \( \lambda \).

Let me compute these.The solutions to the system of equations are:

\[
(x, y, \lambda) = \left(-\frac{3\sqrt{10}}{2}, \frac{\sqrt{10}}{2}, 10\right), \left(-\frac{\sqrt{10}}{2}, -\frac{3\sqrt{10}}{2}, 0\right), \left(\frac{\sqrt{10}}{2}, \frac{3\sqrt{10}}{2}, 0\right), \left(\frac{3\sqrt{10}}{2}, -\frac{\sqrt{10}}{2}, 10\right)
\]

These are the critical points. 

### Step 4: Classify the points
To classify the points as constrained maxima or minima, we evaluate \( f(x, y) \) at each critical point.

Let me compute those values.The values of \( f(x, y) \) at the critical points are:

1. \( f\left(-\frac{3\sqrt{10}}{2}, \frac{\sqrt{10}}{2}\right) = 250 \)
2. \( f\left(-\frac{\sqrt{10}}{2}, -\frac{3\sqrt{10}}{2}\right) = 0 \)
3. \( f\left(\frac{\sqrt{10}}{2}, \frac{3\sqrt{10}}{2}\right) = 0 \)
4. \( f\left(\frac{3\sqrt{10}}{2}, -\frac{\sqrt{10}}{2}\right) = 250 \)

### Conclusion:
- The points \( \left(-\frac{3\sqrt{10}}{2}, \frac{\sqrt{10}}{2}\right) \) and \( \left(\frac{3\sqrt{10}}{2}, -\frac{\sqrt{10}}{2}\right) \) give \( f(x, y) = 250 \), which are points of constrained maxima.
- The points \( \left(-\frac{\sqrt{10}}{2}, -\frac{3\sqrt{10}}{2}\right) \) and \( \left(\frac{\sqrt{10}}{2}, \frac{3\sqrt{10}}{2}\right) \) give \( f(x, y) = 0 \), which are points of constrained minima.

Would you like more details on any step or a breakdown of a particular concept?

Here are some related questions for practice:

1. What is the significance of the Lagrange multiplier \( \lambda \)?
2. How do we determine whether the stationary points are maxima or minima geometrically?
3. How can we extend this method to higher dimensions?
4. What are other methods to solve constrained optimization problems?
5. Can this method be used if the constraint is non-linear?

**Tip**: When solving optimization problems, always check if the constraint forms a boundary on which the extrema occur.

Find the stationary points of the following functions f(x, y) under the constraint g(x, y) = 0, and classify them as points of constrained maximum or points of constrained minimum: f(x, y) = 9x^2 - 6xy + y^2, g(x, y) = x^2 + y^2 - 25.

Solve for the stationary points of the function f(x, y) = 9x^2 - 6xy + y^2 under the constraint g(x, y) = x^2 + y^2 - 25 using Lagrange multipliers. Learn how to classify the points as maxima or minima. This problem covers advanced multivariable calculus concepts and is suitable for college-level students.

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