Question

Consider the function f (x )= -3x^2+18x-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.
c. Identify the function's domain and its range.

a. The function has a maximum or minimum value.

b. The minimum/maximum value is blank. it occurs at x = blank.

c. The domain of f is blank. (Type your answer in interval notation.)

The range of f is blank. (Type your answer in interval notation.)

Answer 1

Let's analyze the function \( f(x) = -3x^2 + 18x - 6 \) step by step.

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

The function is a quadratic function of the form \( ax^2 + bx + c \). In this case, \( a = -3 \), which is less than 0. Since the coefficient of \( x^2 \) (i.e., \( a \)) is negative, the parabola opens downwards. Therefore, the function has a **maximum value**.

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

To find the \( x \)-coordinate of the vertex (which gives the maximum value for downward-opening parabolas), we can use the formula:
\[
x = -\frac{b}{2a}
\]
Here, \( b = 18 \) and \( a = -3 \):

\[
x = -\frac{18}{2(-3)} = -\frac{18}{-6} = 3
\]

Next, we substitute \( x = 3 \) back into the function to find the maximum value:

\[
f(3) = -3(3)^2 + 18(3) - 6
\]
\[
= -3(9) + 54 - 6
\]
\[
= -27 + 54 - 6
\]
\[
= 21
\]

So, the maximum value is **21**, and it occurs at \( x = 3 \).

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

The domain of a quadratic function is all real numbers, since there are no restrictions on \( x \). In interval notation, the domain is:
\[
(-\infty, \infty)
\]

The range of the function can be determined from the maximum value found earlier. Since the function opens downwards (and hence has a maximum), the range starts from negative infinity up to the maximum value of 21. Therefore, the range is:
\[
(-\infty, 21]
\]

### Final Answers:

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

b. The maximum value is **21**. It occurs at \( x = 3 \).

c. The domain of \( f \) is \( (-\infty, \infty) \).

The range of \( f \) is \( (-\infty, 21] \).