Answer :
To analyze the function [tex]\( f(x) = x^5 - 9x^4 + 29x^3 - 45x^2 + 54x - 54 \)[/tex], we can follow these steps:
### Step 1: Find Critical Points
1. Find the Derivative: The first step is to find the derivative of [tex]\( f(x) \)[/tex], which will help us determine the critical points.
[tex]\[
f'(x) = 5x^4 - 36x^3 + 87x^2 - 90x + 54
\][/tex]
2. Solve for Critical Points: Set the derivative equal to zero and solve for [tex]\( x \)[/tex].
[tex]\[
5x^4 - 36x^3 + 87x^2 - 90x + 54 = 0
\][/tex]
Solving this equation will give us the critical points where the function could have local maxima, minima, or points of inflection.
### Step 2: Evaluate the Function at Critical Points
Once we have the critical points, we substitute them back into the original function [tex]\( f(x) \)[/tex] to find the corresponding [tex]\( y \)[/tex]-values. This will help us determine the nature of these critical points.
### Step 3: Analyze Increasing/Decreasing Behavior
To determine where the function is increasing or decreasing:
1. Sign of the Derivative: Check where the derivative [tex]\( f'(x) \)[/tex] is positive (function is increasing) and where it is negative (function is decreasing).
### Step 4: Determine Local Maxima and Minima
By using the first derivative test, we can analyze the behavior at the critical points:
- If [tex]\( f'(x) \)[/tex] changes from positive to negative at a critical point, the function has a local maximum there.
- If [tex]\( f'(x) \)[/tex] changes from negative to positive at a critical point, the function has a local minimum there.
### Optional: Second Derivative Test
For confirmation, the second derivative [tex]\( f''(x) \)[/tex] can be used:
- If [tex]\( f''(x) > 0 \)[/tex] at a critical point, the function has a local minimum.
- If [tex]\( f''(x) < 0 \)[/tex] at a critical point, the function has a local maximum.
This structured approach will help fully analyze the behavior of the function [tex]\( f(x) \)[/tex].
### Step 1: Find Critical Points
1. Find the Derivative: The first step is to find the derivative of [tex]\( f(x) \)[/tex], which will help us determine the critical points.
[tex]\[
f'(x) = 5x^4 - 36x^3 + 87x^2 - 90x + 54
\][/tex]
2. Solve for Critical Points: Set the derivative equal to zero and solve for [tex]\( x \)[/tex].
[tex]\[
5x^4 - 36x^3 + 87x^2 - 90x + 54 = 0
\][/tex]
Solving this equation will give us the critical points where the function could have local maxima, minima, or points of inflection.
### Step 2: Evaluate the Function at Critical Points
Once we have the critical points, we substitute them back into the original function [tex]\( f(x) \)[/tex] to find the corresponding [tex]\( y \)[/tex]-values. This will help us determine the nature of these critical points.
### Step 3: Analyze Increasing/Decreasing Behavior
To determine where the function is increasing or decreasing:
1. Sign of the Derivative: Check where the derivative [tex]\( f'(x) \)[/tex] is positive (function is increasing) and where it is negative (function is decreasing).
### Step 4: Determine Local Maxima and Minima
By using the first derivative test, we can analyze the behavior at the critical points:
- If [tex]\( f'(x) \)[/tex] changes from positive to negative at a critical point, the function has a local maximum there.
- If [tex]\( f'(x) \)[/tex] changes from negative to positive at a critical point, the function has a local minimum there.
### Optional: Second Derivative Test
For confirmation, the second derivative [tex]\( f''(x) \)[/tex] can be used:
- If [tex]\( f''(x) > 0 \)[/tex] at a critical point, the function has a local minimum.
- If [tex]\( f''(x) < 0 \)[/tex] at a critical point, the function has a local maximum.
This structured approach will help fully analyze the behavior of the function [tex]\( f(x) \)[/tex].
