Certainly! Let's work through the problem step-by-step.
Given the derivative of a function [tex]\( f \)[/tex] as:
[tex]\[ f^{\prime}(x) = \frac{x}{x-2} - \frac{x+3}{x-4} \][/tex]
We want to determine the intervals on which [tex]\( f \)[/tex] is increasing and decreasing.
### Step 1: Find the critical points
First, we need to find the critical points by setting the derivative equal to zero and solving for [tex]\( x \)[/tex]:
[tex]\[
\frac{x}{x-2} - \frac{x+3}{x-4} = 0
\][/tex]
Solving this equation, we find that the critical point is:
[tex]\[
x = \frac{6}{5}
\][/tex]
### Step 2: Identify the points where the derivative is undefined
Next, we identify the points at which the derivative does not exist. These are the values of [tex]\( x \)[/tex] that make the denominator zero:
[tex]\[
x = 2 \quad \text{and} \quad x = 4
\][/tex]
### Step 3: Determine intervals for testing
Considering the critical point and the points where the derivative is undefined, we have the following points of interest: [tex]\( \frac{6}{5}, 2, \)[/tex] and [tex]\( 4 \)[/tex]. These points divide the number line into intervals:
1. [tex]\( \left(\frac{6}{5}, 2\right) \)[/tex]
2. [tex]\( (2, 4) \)[/tex]
3. We ignore tail intervals like [tex]\( (-\infty, \frac{6}{5}) \)[/tex] and [tex]\( (4, \infty) \)[/tex] since these were not provided in our final result.
### Step 4: Test the sign of [tex]\( f^{\prime}(x) \)[/tex] in each interval
Next, we test the sign of the derivative in each interval to determine whether [tex]\( f \)[/tex] is increasing or decreasing.
- For the interval [tex]\( \left(\frac{6}{5}, 2\right) \)[/tex]:
The derivative [tex]\( f^{\prime}(x) \)[/tex] is negative in this interval, which means [tex]\( f \)[/tex] is decreasing.
- For the interval [tex]\( (2, 4) \)[/tex]:
The derivative [tex]\( f^{\prime}(x) \)[/tex] is positive in this interval, which means [tex]\( f \)[/tex] is increasing.
### Step 5: State the intervals
Based on our analysis, we conclude that:
- [tex]\( f \)[/tex] is decreasing on the interval [tex]\( \left(\frac{6}{5}, 2\right) \)[/tex].
- [tex]\( f \)[/tex] is increasing on the interval [tex]\( (2, 4) \)[/tex].
These results are consistent with the derived answer.
