The graph and equation of the function f are given.

**a.** Use the graph to find any values at which f has a relative​ maximum, and use the equation to calculate the relative maximum for each value.

**b.** Use the graph to find any values at which f has a relative​ minimum, and use the equation to calculate the relative minimum for each value.

...

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Part 1

f left parenthesis x right parenthesis equals 2 x cubed minus 6 x squared minus 18 x plus 1f(x)=2x3−6x2−18x+1

A coordinate system has a horizontal axis labeled from negative 5 to 5 in increments of 1 and a vertical axis labeled from negative 70 to 70 in increments of 10. From left to right, a curve rises and is horizontal when passing through the horizontal axis value negative 1, then falls and is horizontal when passing through the horizontal axis value 3, and then rises.

​[negative 5−5​,55​,11​] 

by 

​[negative 70−70​,7070​,1010​]

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Part 1

**a.** Select the correct choice below​ and, if​ necessary, fill in the answer boxes to complete your choice.

A.

The function f has​ (a) relative​ maxima(maximum) at 
 
enter your response here 

and the relative​ maxima(maximum) are(is) 
 
enter your response here.

​(Use a comma to separate answers as​ needed.)

B.

The function f has no relative maxima.

Let's analyze the given function \( f(x) = 2x^3 - 6x^2 - 18x + 1 \) and use the provided graph description to identify the relative maximum and minimum points.

### (a) Relative Maximum:
From the description:
- The graph rises, becomes horizontal (indicating a potential maximum), and then falls at \( x = -1 \).

**Steps to Calculate the Relative Maximum:**

1. **Find the derivative of the function \( f(x) \) to identify critical points:**

   \[
   f'(x) = \frac{d}{dx} \left( 2x^3 - 6x^2 - 18x + 1 \right)
   \]
   \[
   f'(x) = 6x^2 - 12x - 18
   \]

2. **Set the derivative equal to zero to find critical points:**

   \[
   6x^2 - 12x - 18 = 0
   \]
   \[
   x^2 - 2x - 3 = 0
   \]
   \[
   (x - 3)(x + 1) = 0
   \]
   The critical points are \( x = 3 \) and \( x = -1 \).

3. **Use the second derivative test to determine whether these critical points correspond to maxima or minima:**

   \[
   f''(x) = \frac{d}{dx}(6x^2 - 12x - 18) = 12x - 12
   \]

   - At \( x = -1 \): 
     \[
     f''(-1) = 12(-1) - 12 = -24 \quad \text{(Since \( f''(-1) < 0 \), \( x = -1 \) is a relative maximum.)}
     \]

4. **Calculate the value of the function at \( x = -1 \) to find the relative maximum:**
   \[
   f(-1) = 2(-1)^3 - 6(-1)^2 - 18(-1) + 1
   \]
   \[
   f(-1) = -2 - 6 + 18 + 1 = 11
   \]

### Answer:
**A.** The function \( f \) has a relative maximum at \( x = -1 \), and the value of this relative maximum is 11.

---

### (b) Relative Minimum:
From the description:
- The graph falls, becomes horizontal (indicating a potential minimum), and then rises at \( x = 3 \).

**Steps to Calculate the Relative Minimum:**

1. **Second derivative test at \( x = 3 \):**
   \[
   f''(3) = 12(3) - 12 = 36 - 12 = 24 \quad \text{(Since \( f''(3) > 0 \), \( x = 3 \) is a relative minimum.)}
   \]

2. **Calculate the value of the function at \( x = 3 \) to find the relative minimum:**
   \[
   f(3) = 2(3)^3 - 6(3)^2 - 18(3) + 1
   \]
   \[
   f(3) = 2(27) - 6(9) - 54 + 1 = 54 - 54 - 54 + 1 = -53
   \]

### Answer:
The function \( f \) has a relative minimum at \( x = 3 \), and the value of this relative minimum is \(-53\).

---

Do you have any further questions, or would you like more details on this explanation?

### Related Questions:
1. How can you use the first derivative test to identify relative maxima and minima?
2. What role does the second derivative play in determining the concavity of a function?
3. How can you find the inflection points of a cubic function?
4. What is the significance of the discriminant in determining the nature of critical points?
5. How would you sketch the graph of a cubic function based on its derivatives?

### Tip:
For polynomial functions, critical points are found where the derivative equals zero. To classify these points, use the second derivative test: if positive, it's a minimum; if negative, it's a maximum.

Learn how to identify relative extrema of a function using graphical analysis and calculations. This problem involves analyzing the function f(x) = 2x^3 - 6x^2 - 18x + 1 to find its relative maximum and minimum points, using the provided graph description. Suitable for high school students studying calculus and graphical analysis.

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