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Part 1

Find ​a) any critical values and ​b) any relative extrema.

​f(x)equals

x squared plus 8 x plus 18

To find the critical values and relative extrema of the function

\[
f(x) = x^2 + 8x + 18
\]

we'll follow these steps:

### Step 1: Find the First Derivative
The first derivative, \( f'(x) \), will help us locate critical values by setting it equal to zero.

\[
f'(x) = 2x + 8
\]

### Step 2: Solve for Critical Values
To find the critical points, set \( f'(x) = 0 \):

\[
2x + 8 = 0
\]

Solve for \( x \):

\[
x = -4
\]

Thus, \( x = -4 \) is a critical value.

### Step 3: Determine if the Critical Point is a Relative Minimum or Maximum
To classify the critical point, we can use the second derivative test or analyze the nature of \( f(x) \) at \( x = -4 \).

#### Second Derivative Test
1. Find the second derivative of \( f(x) \):
   \[
   f''(x) = 2
   \]
2. Since \( f''(x) = 2 \) (which is positive), the critical point \( x = -4 \) is a **relative minimum**.

### Step 4: Calculate the Value of \( f(x) \) at \( x = -4 \)
To find the relative minimum value:

\[
f(-4) = (-4)^2 + 8(-4) + 18 = 16 - 32 + 18 = 2
\]

### Summary
- **a) Critical value**: \( x = -4 \)
- **b) Relative minimum**: \( f(-4) = 2 \) at \( x = -4 \)

Would you like more details or have any questions?

### Related Questions:
1. How do you apply the first and second derivative tests to other functions?
2. Can this approach work for functions with more complex terms, like trigonometric functions?
3. What are absolute extrema, and how do they differ from relative extrema?
4. How would the analysis change if the function had a negative coefficient for \( x^2 \)?
5. Can you have more than one critical point in a polynomial function?

**Tip:** To confirm extrema classification, graphing the function can offer a visual verification of minimum and maximum points.

Find a) any critical values and b) any relative extrema. f(x) = x^2 + 8x + 18

Explore how to find critical values and relative extrema for the function f(x) = x^2 + 8x + 18. This calculus problem includes key steps such as finding the first and second derivatives and applying the second derivative test. Ideal for high school students in Grades 11-12.

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