`f(x) = x/(x - 5)` Find the critical numbers, open intervals on which the function is increasing or decreasing, apply first derivative test to identify all relative extrema.

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Chapter 3, 3.3 - Problem 34 - Calculus of a Single Variable (10th Edition, Ron Larson).
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mathace | (Level 3) Assistant Educator

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Given: `f(x)=x/(x-5)`

Find the critical numbers by setting the first derivative equal to zero and solving for the x value(x)

`f'(x)=[(x-5)(1)-x(1)]/(x-5)^2=0`

`x-5-x=0`

`-5=0`

No solution.

A critical number cannot be obtained using the first derivative. Critical values also exist where f(x) is not defined. Therefore there will be a critical number at x=5.

If f'(x)>0 the function is increasing in the interval.

If f'(x)<0 the function is decreasing in the interval.

Choose a value less than 5.

f'(4)=-5 Since f'(4)<0 the function is decreasing in the interval

(-`oo,5).`

Choose a value greater than 5. 

f'(6)=-5 Since f'(6)<0 the function is decreasing in the interval (5, `oo).`

Because the direction of the function did not change, there are NO relative extrema.

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