Consider the function $$ \mathrm{f}(\mathrm{x})=-2 \mathrm{x}^{2}+16 \mathrm{x}-6 $$. a. Determine, without graphing, whether the function has a minimum value or a maximum value. b. Find the minimum or maximum value and determine where it occurs. Identify the function's domain and its range. c. The function has a b. The minimum/maximum value is $$ \square $$. It occurs at $$ \mathrm{x}=\square $$. 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 $$ \mathrm{x}^{2} $$ (which is $$-2$$) is negative, the parabola opens downward. Therefore, the function has a maximum value.

b. The vertex of a quadratic function $$ \mathrm{f}(\mathrm{x}) = ax^2 + bx + c $$ occurs at 
$$
\mathrm{x} = -\frac{b}{2a}.
$$
For this function, $$ a = -2 $$ and $$ b = 16 $$. Thus,
$$
\mathrm{x} = -\frac{16}{2(-2)} = 4.
$$
Substitute $$ \mathrm{x} = 4 $$ into the function to find the maximum value:
$$
\mathrm{f}(4) = -2(4)^2 + 16(4) - 6 = -2(16) + 64 - 6 = -32 + 64 - 6 = 26.
$$

c. The answers are:  
b. The maximum value is $$ 26 $$. It occurs at $$ \mathrm{x}=4 $$.  
c. The domain of $$ \mathrm{f} $$ is $$(-\infty, \infty)$$.  
The range of $$ \mathrm{f} $$ is $$(-\infty, 26]$$.
a. The function has a maximum value because the coefficient of $$ x^2 $$ is negative. b. The maximum value is 26, which occurs at $$ x = 4 $$. c. The domain of $$ f $$ is all real numbers, written as $$ (-\infty, \infty) $$. The range of $$ f $$ is all real numbers less than or equal to 26, written as $$ (-\infty, 26] $$.

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