Find the stationary point for f(x1,x2)
.

0.6, 0.4
 

−0.6, −0.4
 

0.2, −0.6
 

0.2, 0.6

To find the stationary point of a function \( f(x_1, x_2) \), we need to solve the system of equations where the gradient \( \nabla f \) is equal to zero. In other words, we solve for \( x_1 \) and \( x_2 \) by setting the partial derivatives to zero.

Given:
\[
f(x_1, x_2) = 2x_1^2 + 3x_1x_2 + 3x_2^2 + x_1 + 3x_2
\]
We previously found the gradient:
\[
\nabla f = \left[ 4x_1 + 3x_2 + 1, 3x_1 + 6x_2 + 3 \right]
\]

### Set the gradient equal to zero:
1. \( 4x_1 + 3x_2 + 1 = 0 \)
2. \( 3x_1 + 6x_2 + 3 = 0 \)

### Solve the system of equations:

Let's solve this system step-by-step.

1. **From equation (2):**
   \[
   3x_1 + 6x_2 + 3 = 0 \quad \text{or} \quad x_1 + 2x_2 + 1 = 0
   \]
   \[
   x_1 = -2x_2 - 1
   \]

2. **Substitute \( x_1 = -2x_2 - 1 \) into equation (1):**
   \[
   4(-2x_2 - 1) + 3x_2 + 1 = 0
   \]
   \[
   -8x_2 - 4 + 3x_2 + 1 = 0
   \]
   \[
   -5x_2 - 3 = 0
   \]
   \[
   x_2 = -\frac{3}{5} = -0.6
   \]

3. **Substitute \( x_2 = -0.6 \) back into \( x_1 = -2x_2 - 1 \):**
   \[
   x_1 = -2(-0.6) - 1 = 1.2 - 1 = 0.2
   \]

### Conclusion:
The stationary point is \( (x_1, x_2) = (0.2, -0.6) \).

Thus, the correct answer is:
\[
\boxed{0.2, -0.6}
\]

Do you want further clarification on this? Here are some related questions:

1. What does a stationary point represent geometrically?
2. How do we classify a stationary point as a minimum, maximum, or saddle point?
3. What role does the Hessian matrix play in determining the nature of a stationary point?
4. How do we check if a stationary point is a local or global extremum?
5. Can there be more than one stationary point for a given function?
6. How do we extend this approach to functions with more variables?
7. What is the significance of the second derivative test in multivariable functions?
8. How can we use Lagrange multipliers to find stationary points with constraints?

**Tip:** Stationary points are found where the gradient is zero, but always check the second derivatives to classify the nature of the point.

Learn how to find the stationary point for the function f(x1,x2) using multivariable calculus techniques. This problem involves calculating gradients, solving systems of equations, and understanding the concept of stationary points. Suitable for advanced college-level mathematics.

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