 Find the relative extreme using nth order derivative test for the function 𝑓(𝑥) =
 (5 −𝑥) 

The function given is:

\[
f(x) = 5 - x
\]

To find the relative extrema (relative maximum or minimum) using the nth-order derivative test, we follow these steps:

### Step 1: Find the first derivative, \( f'(x) \)
\[
f'(x) = \frac{d}{dx}(5 - x) = -1
\]

### Step 2: Find the critical points
To find critical points, set \( f'(x) = 0 \):
\[
-1 = 0
\]
This equation has no solution. Therefore, there are no critical points for the function.

### Step 3: Analyze the behavior of the function
Since the first derivative is a constant \( -1 \), the function is decreasing for all values of \( x \). Because there are no critical points, there are no relative extrema.

Thus, the function \( f(x) = 5 - x \) is a straight line with a constant negative slope, which means it does not have any relative maxima or minima.

### Conclusion:
There are no relative extrema for the function \( f(x) = 5 - x \).

Would you like more details or have any questions?

### Related Questions:
1. How do you determine relative extrema for quadratic functions?
2. What is the second derivative test, and how does it work?
3. Can the nth-order derivative test help in finding inflection points?
4. How do you find global extrema on a closed interval?
5. What is the geometric interpretation of the nth-order derivative test?

### Tip:
For linear functions like \( f(x) = 5 - x \), there are no relative extrema since the function always increases or decreases.

Find the relative extreme using nth order derivative test for the function f(x) = 5 - x.

Learn how to find the relative extrema of the linear function f(x) = 5 - x using the nth-order derivative test. This problem demonstrates the process of analyzing derivatives and understanding the behavior of linear functions. Suitable for students in Grades 11-12.

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