`g(x) = x^2 - 2x - 8` Identify the open intervals on which the function is increasing or decreasing.

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Chapter 3, 3.3 - Problem 9 - Calculus of a Single Variable (10th Edition, Ron Larson).
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mathace | (Level 3) Assistant Educator

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Given: `g(x)=x^2-2x-8`

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

`g'(x)=2x-2=0`

`2x=2`

`x=1`

The critical value is x=1.

If g'(x)>0 the function will increase over an interval.

If g'(x)<0 the function will decrease over an interval.

Choose an x value less than 1.

g'(0)=-2  Since g'(0)<0 the function will decrease in the interval (-`oo, 1).`

Choose an x value greater than 2.

g'(2)=2  Since g'(2)>0 the function will increase in the interval (1, `oo).`

Because the function changed direction from decreasing to increasing a relative minimum will exist. The relative minimum will occur at the point (1, -9).

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