Using the completing-the-square method, find the vertex of the function f(x) = −3x2 + 6x − 2 and indicate whether it is a minimum or a maximum and at what point.

To find the vertex of the quadratic function \( f(x) = -3x^2 + 6x - 2 \) using the completing-the-square method, we follow these steps:

1. **Factor out the coefficient of \( x^2 \)** from the first two terms:
\[
f(x) = -3(x^2 - 2x) - 2
\]

2. **Complete the square inside the parentheses**. To do this, take half of the coefficient of \( x \) (which is \(-2\)), square it, and add and subtract that value inside the parentheses. Half of \(-2\) is \(-1\), and squaring it gives us \(1\):
\[
f(x) = -3(x^2 - 2x + 1 - 1) - 2
\]
This simplifies to:
\[
f(x) = -3((x - 1)^2 - 1) - 2
\]

3. **Distribute the \(-3\)**:
\[
f(x) = -3(x - 1)^2 + 3 - 2
\]
So, we have:
\[
f(x) = -3(x - 1)^2 + 1
\]

4. **Identify the vertex**. The vertex form of a quadratic function is given by \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex. From the equation, we see that:
\[
h = 1 \quad \text{and} \quad k = 1
\]
Thus, the vertex is \((1, 1)\).

5. **Determine whether it's a maximum or minimum**. Since the coefficient of the squared term \(-3\) is negative, the parabola opens downward, indicating that the vertex represents a maximum point.

Therefore, the vertex of the function \( f(x) = -3x^2 + 6x - 2 \) is a maximum at the point \((1, 1)\).

The final answer is:
**Maximum at (1, 1)**.