1. Suppose that the derivative of a function [tex][tex]$f$[/tex][/tex] is given by [tex][tex]$f^{\prime}(x)=\frac{x}{x-2}-\frac{x+3}{x-4}$[/tex][/tex]. Find the intervals on which [tex][tex]$f$[/tex][/tex] is increasing and decreasing.



Answer :

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.