Consider the function $$ f(x)=-3 x^{2}+18 x-7 $$. a. Determine, without graphing, whether the function has a minimum value b. Find the minimum or maximum value and determine where it occurs. c. Identify the function's domain and its range. a. The function has a $$ \square $$ value. b. The minimum/maximum value is $$ \square $$. It occurs at $$ x=\square $$. c. The domain of f is $$ \square $$. (Type your answer in interval notation.) The range of f is $$ \square $$. (Type your answer in interval notation.)

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a. Since the coefficient of $$ x^2 $$ is $$-3$$ (which is negative), the parabola opens downward. Therefore, the function does not have a minimum value; it has a maximum value.

b. The $$ x $$-coordinate of the vertex (where the maximum occurs) is given by  
$$
x = -\frac{b}{2a} = -\frac{18}{2(-3)} = 3.
$$  
Substitute $$ x=3 $$ into the function to find the maximum value:  
$$
f(3) = -3 (3)^2 + 18 (3) - 7 = -3 \times 9 + 54 - 7 = -27 + 54 - 7 = 20.
$$  
Thus, the maximum value is $$ 20 $$ and it occurs at $$ x = 3 $$.

c. The domain of $$ f $$ is all real numbers, which in interval notation is  
$$
(-\infty, \infty).
$$  
Since the parabola opens downward and the maximum value is $$ 20 $$, the range is all values less than or equal to $$ 20 $$. In interval notation, the range is  
$$
(-\infty, 20].
$$

a. The function has a **maximum** value.  
b. The maximum value is $$ 20 $$. It occurs at $$ x = 3 $$.  
c. The domain of $$ f $$ is $$ (-\infty, \infty) $$.  
The range of $$ f $$ is $$ (-\infty, 20] $$.
a. The function has a **maximum** value. b. The maximum value is $$ 20 $$. It occurs at $$ x = 3 $$. c. The domain of $$ f $$ is $$ (-\infty, \infty) $$. The range of $$ f $$ is $$ (-\infty, 20] $$.

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