Consider the function $$ f(x)=-2 x^{2}+20 x-5 $$. 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. c. Identify the function's domain and its range. a. The function has a

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To analyze the function $$ f(x) = -2x^2 + 20x - 5 $$, we can follow the steps outlined in your question.

### a. Determine whether the function has a minimum value or a maximum value.

The function $$ f(x) $$ is a quadratic function of the form $$ ax^2 + bx + c $$, where:
- $$ a = -2 $$
- $$ b = 20 $$
- $$ c = -5 $$

Since the coefficient $$ a $$ is negative ($$ a < 0 $$), the parabola opens downwards. This means that the function has a **maximum value**.

### b. Find the maximum value and determine where it occurs.

The maximum (or minimum) value of a quadratic function occurs at the vertex, which can be found using the formula for the x-coordinate of the vertex:
$$
x = -\frac{b}{2a}
$$
Substituting the values of $$ a $$ and $$ b $$:
$$
x = -\frac{20}{2 \cdot -2} = -\frac{20}{-4} = 5
$$

Now, we can find the maximum value by substituting $$ x = 5 $$ back into the function:
$$
f(5) = -2(5^2) + 20(5) - 5
$$
Calculating this step-by-step:
$$
f(5) = -2(25) + 100 - 5
$$
$$
f(5) = -50 + 100 - 5
$$
$$
f(5) = 45
$$

Thus, the maximum value of the function is $$ 45 $$, and it occurs at $$ x = 5 $$.

### c. Identify the function's domain and its range.

- **Domain**: The domain of a quadratic function is all real numbers, so:
$$
\text{Domain: } (-\infty, \infty)
$$

- **Range**: Since the function has a maximum value of $$ 45 $$ and opens downwards, the range is all values less than or equal to $$ 45 $$:
$$
\text{Range: } (-\infty, 45]
$$

### Summary of Results:
a. The function has a **maximum value**.  
b. The maximum value is **45**, occurring at $$ x = 5 $$.  
c. The domain is $$ (-\infty, \infty) $$ and the range is $$ (-\infty, 45] $$.
a. The function has a maximum value. b. The maximum value is 45, occurring at $$ x = 5 $$. c. The domain is $$ (-\infty, \infty) $$ and the range is $$ (-\infty, 45] $$.

Consider the function .
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.
c. Identify the function’s domain and its range.
a. The function has a

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Consider the function . 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. c. Identify the function’s domain and its range. a. The function has a