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We are given the function \( f: \mathbb{R}^2 \to \mathbb{R} \) defined as follows:

\[
f(x, y) = x^2 + 2y^2 - 3xy
\]

The task is to compute the sum of the partial derivatives at the point \( (0, 0) \), specifically:

\[
f_x(0,0) + f_y(0,0) + f_{xy}(0,0)
\]

### Step 1: Compute \( f_x(x, y) \) (Partial derivative with respect to \( x \))
The partial derivative of \( f(x, y) \) with respect to \( x \) is:
\[
f_x(x, y) = \frac{\partial}{\partial x}(x^2 + 2y^2 - 3xy) = 2x - 3y
\]
At \( (0, 0) \):
\[
f_x(0, 0) = 2(0) - 3(0) = 0
\]

### Step 2: Compute \( f_y(x, y) \) (Partial derivative with respect to \( y \))
The partial derivative of \( f(x, y) \) with respect to \( y \) is:
\[
f_y(x, y) = \frac{\partial}{\partial y}(x^2 + 2y^2 - 3xy) = 4y - 3x
\]
At \( (0, 0) \):
\[
f_y(0, 0) = 4(0) - 3(0) = 0
\]

### Step 3: Compute \( f_{xy}(x, y) \) (Mixed partial derivative with respect to \( x \) and \( y \))
First, we compute \( f_{xy}(x, y) \) by differentiating \( f_x(x, y) = 2x - 3y \) with respect to \( y \):
\[
f_{xy}(x, y) = \frac{\partial}{\partial y}(2x - 3y) = -3
\]
At \( (0, 0) \):
\[
f_{xy}(0, 0) = -3
\]

### Step 4: Sum the derivatives
Now we sum the derivatives:
\[
f_x(0, 0) + f_y(0, 0) + f_{xy}(0, 0) = 0 + 0 - 3 = -3
\]

Thus, the value of \( f_x(0,0) + f_y(0,0) + f_{xy}(0,0) \) is \( \boxed{-3} \).

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Would you like further details on any of the steps? Here are some related questions for further exploration:

1. How do you compute the second-order mixed partial derivatives for a function of two variables?
2. What conditions must a function satisfy for \( f_{xy} = f_{yx} \)?
3. How does the chain rule extend to functions of several variables?
4. Can a function of two variables have discontinuous partial derivatives?
5. How does the Hessian matrix help in understanding second-order partial derivatives?

**Tip:** When computing partial derivatives, always check if mixed derivatives commute, i.e., \( f_{xy} = f_{yx} \), using Clairaut's theorem.

Let f: R^2 -> R be defined by f(x, y) = x^2 + 2y^2 - 3xy. Find the value of f_x(0,0) + f_y(0,0) + f_xy(0,0).

Compute the partial derivatives of the function f(x, y) = x^2 + 2y^2 - 3xy at (0,0). Learn how to find f_x, f_y, and the mixed derivative f_xy, along with the final sum of these derivatives. This problem demonstrates the application of multivariable calculus techniques, such as Clairaut's theorem, and is suitable for university-level students.

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