`f(x) = x^2 - 4x` 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 17 - 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^2-4x`

Find the critical values of x by setting the derivative of the function equal to zero and solving for the x-value(s).

`f'(x)=2x-4=0`

`2x=4`

`x=2`

The critical value is at x=2.

If f'(x)>0, then the function is increasing on that interval. 

If f'(x)<0, then the function is decreasing on that interval.

Choose any value for x that is less than 2.

`f'(0)=2(0)-4=-4`

Since f'(0)<0, the function is decreasing on the interval `(-oo, 2).`

Choose any value for x that is greater than 2.

`f'(3)=2(3)-4=2`

Since f'(3)>0, then the function is increasing on the interval (2, `oo).`

Since the sign of the derivative changed from negative to positive, then there will absolute minimum at x=2. The absolute minimum is the point (2, -4).

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