Question
How to apply the First Derivative Test to a quartic function to find critical points and determine intervals of increase and decrease?
Answers
Answered by
GPT-4o mini
To apply the First Derivative Test to a quartic function for finding critical points and determining the intervals of increase and decrease, follow these steps:
### Step 1: Find the First Derivative
Given a quartic function \( f(x) = ax^4 + bx^3 + cx^2 + dx + e \) where \( a \), \( b \), \( c \), \( d \), and \( e \) are constants, calculate the first derivative \( f'(x) \):
\[
f'(x) = 4ax^3 + 3bx^2 + 2cx + d
\]
### Step 2: Find Critical Points
Set the first derivative equal to zero to find the critical points:
\[
f'(x) = 0
\]
Solve this equation for \( x \). Since \( f'(x) \) is a cubic polynomial, it can have up to three real roots. These roots are your critical points.
### Step 3: Test Intervals
Once you have the critical points, divide the real line into intervals based on these points. For each interval, you will select a test point to determine the sign of the derivative in that interval.
### Step 4: Analyze the Sign of the First Derivative
Choose a test point from each interval. For each test point \( x_0 \), evaluate \( f'(x_0) \):
- If \( f'(x_0) > 0 \), then \( f(x) \) is **increasing** on that interval.
- If \( f'(x_0) < 0 \), then \( f(x) \) is **decreasing** on that interval.
### Step 5: Apply the First Derivative Test
Using the information gathered from evaluating the sign of the derivative at each critical point:
- If \( f'(x) \) changes from positive to negative at a critical point \( x = c \), then \( f(c) \) is a **local maximum**.
- If \( f'(x) \) changes from negative to positive at \( x = c \), then \( f(c) \) is a **local minimum**.
- If \( f'(x) \) does not change sign (i.e., stays positive or negative), then \( f(c) \) is neither a maximum nor a minimum (this could be an inflection point, for example).
### Example
Consider a simple quartic function for illustration:
\[
f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1
\]
**Step 1:** Calculate the first derivative:
\[
f'(x) = 4x^3 - 12x^2 + 12x - 4
\]
**Step 2:** Set \( f'(x) = 0 \) and solve for \( x \):
This cubic equation may be solved through methods such as factoring, synthetic division, or numerical methods.
**Step 3:** Determine critical points, say you find \( x = 1, 2, 3 \) as critical points.
**Step 4:** Test intervals:
- Choose test points (e.g., \( x = 0, 1.5, 2.5, 4 \)).
- Calculate \( f'(0) \), \( f'(1.5) \), \( f'(2.5) \), and \( f'(4) \), and note their signs.
**Step 5:** Analyze results:
- For a point from decreasing to increasing, you have a minimum and vice versa for a maximum.
This systematic approach allows you to identify the behavior of the quartic function on specified intervals based on its critical points.
### Step 1: Find the First Derivative
Given a quartic function \( f(x) = ax^4 + bx^3 + cx^2 + dx + e \) where \( a \), \( b \), \( c \), \( d \), and \( e \) are constants, calculate the first derivative \( f'(x) \):
\[
f'(x) = 4ax^3 + 3bx^2 + 2cx + d
\]
### Step 2: Find Critical Points
Set the first derivative equal to zero to find the critical points:
\[
f'(x) = 0
\]
Solve this equation for \( x \). Since \( f'(x) \) is a cubic polynomial, it can have up to three real roots. These roots are your critical points.
### Step 3: Test Intervals
Once you have the critical points, divide the real line into intervals based on these points. For each interval, you will select a test point to determine the sign of the derivative in that interval.
### Step 4: Analyze the Sign of the First Derivative
Choose a test point from each interval. For each test point \( x_0 \), evaluate \( f'(x_0) \):
- If \( f'(x_0) > 0 \), then \( f(x) \) is **increasing** on that interval.
- If \( f'(x_0) < 0 \), then \( f(x) \) is **decreasing** on that interval.
### Step 5: Apply the First Derivative Test
Using the information gathered from evaluating the sign of the derivative at each critical point:
- If \( f'(x) \) changes from positive to negative at a critical point \( x = c \), then \( f(c) \) is a **local maximum**.
- If \( f'(x) \) changes from negative to positive at \( x = c \), then \( f(c) \) is a **local minimum**.
- If \( f'(x) \) does not change sign (i.e., stays positive or negative), then \( f(c) \) is neither a maximum nor a minimum (this could be an inflection point, for example).
### Example
Consider a simple quartic function for illustration:
\[
f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1
\]
**Step 1:** Calculate the first derivative:
\[
f'(x) = 4x^3 - 12x^2 + 12x - 4
\]
**Step 2:** Set \( f'(x) = 0 \) and solve for \( x \):
This cubic equation may be solved through methods such as factoring, synthetic division, or numerical methods.
**Step 3:** Determine critical points, say you find \( x = 1, 2, 3 \) as critical points.
**Step 4:** Test intervals:
- Choose test points (e.g., \( x = 0, 1.5, 2.5, 4 \)).
- Calculate \( f'(0) \), \( f'(1.5) \), \( f'(2.5) \), and \( f'(4) \), and note their signs.
**Step 5:** Analyze results:
- For a point from decreasing to increasing, you have a minimum and vice versa for a maximum.
This systematic approach allows you to identify the behavior of the quartic function on specified intervals based on its critical points.