$$ 1 \leftarrow $$ Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find the value. $$ f(x)=-3 x^{2}-6 x $$ The quadratic function has a maximum value. The value is $$ \square $$.

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To determine whether the given quadratic function $$ f(x) = -3x^2 - 6x $$ has a maximum or minimum value, we can analyze the coefficient of the $$ x^2 $$ term.

1. **Identify the coefficient of $$ x^2 $$**:
   - In the function $$ f(x) = -3x^2 - 6x $$, the coefficient of $$ x^2 $$ is $$ -3 $$.

2. **Determine the nature of the quadratic**:
   - If the coefficient of $$ x^2 $$ is positive, the parabola opens upwards, and the function has a minimum value.
   - If the coefficient of $$ x^2 $$ is negative, the parabola opens downwards, and the function has a maximum value.
   - Since $$ -3 < 0 $$, the function has a maximum value.

3. **Find the maximum value**:
   - The maximum value of a quadratic function in the form $$ ax^2 + bx + c $$ occurs at the vertex, which can be found using the formula $$ x = -\frac{b}{2a} $$.
   - Here, $$ a = -3 $$ and $$ b = -6 $$.

Now, let's calculate the $$ x $$-coordinate of the vertex:

$$
x = -\frac{-6}{2 \cdot -3} = \frac{6}{-6} = -1
$$

4. **Substitute $$ x = -1 $$ back into the function to find the maximum value**:

$$
f(-1) = -3(-1)^2 - 6(-1) = -3(1) + 6 = -3 + 6 = 3
$$

Thus, the quadratic function has a maximum value of $$ 3 $$.

The final answer is:
- The quadratic function has a maximum value.
- The value is $$ 3 $$.
The quadratic function $$ f(x) = -3x^2 - 6x $$ has a maximum value of $$ 3 $$.

\( 1 \leftarrow \) Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find the value. \( f(x)=-3 x^{2}-6 x \) The quadratic function has a maximum value. The value is \( \square \).

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