Question

Consider the function f left parenthesis x right parenthesis equals negative 3 x squared plus 6 x minus 2.
a. nbsp
​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.
Question content area bottom
Part 1
a. The function has a
▼
maximum
minimum
value.
Part 2
b. The​ minimum/maximum value is
  
enter your response here. It occurs at xequals
  
enter your response here.
Part 3
c. The domain of f is
  
enter your response here. ​(Type your answer in interval​ notation.)
Part 4
The range of f is
  
enter your response here. ​(Type your answer in interval​ notation.)

Answer 1

To analyze the function \( f(x) = -3x^2 + 6x - 2 \), let's go through the parts one by one.

### Part 1:
**Determine whether the function has a minimum value or a maximum value.**

The function is a quadratic function of the form \( ax^2 + bx + c \). The coefficient of \( x^2 \) is \( a = -3 \), which is negative. Therefore, the parabola opens downwards, meaning the function has a maximum value.

**Answer:**
The function has a **maximum** value.

### Part 2:
**Find the maximum value and determine where it occurs.**

The x-coordinate of the vertex of a quadratic function is given by the formula:

\[
x = -\frac{b}{2a}
\]

Here, \( a = -3 \) and \( b = 6 \):

\[
x = -\frac{6}{2(-3)} = -\frac{6}{-6} = 1
\]

Now, we calculate the maximum value by substituting \( x = 1 \) back into the function:

\[
f(1) = -3(1)^2 + 6(1) - 2
\]
\[
= -3 + 6 - 2
\]
\[
= 1
\]

**Answer:**
The maximum value is **1**. It occurs at \( x = 1 \).

### Part 3:
**Identify the function's domain.**

Since \( f(x) \) is a polynomial function, its domain is all real numbers.

**Answer:**
The domain of \( f \) is \( (-\infty, \infty) \).

### Part 4:
**Identify the range of the function.**

Since the function has a maximum value of 1 and it opens downward, the range will be all real numbers less than or equal to the maximum value.

**Answer:**
The range of \( f \) is \( (-\infty, 1] \).

### Summary of Answers:
- Part 1: The function has a **maximum** value.
- Part 2: The maximum value is **1**. It occurs at \( x = 1 \).
- Part 3: The domain of \( f \) is \( (-\infty, \infty) \).
- Part 4: The range of \( f \) is \( (-\infty, 1] \).