`f(x) = x^2 + 6x + 10` 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 18 - 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+6x+10`

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

`f'(x)=2x+6=0`

`f'(x)=2x=-6`

`x=-3` 

The critical value for the first derivative is x=-3

If f'(x)>0, the function will increase in the interval.

If f'(x)<0, the function will decrease in the interval.

Choose a value for x that is less than -3.

f'(-4)=-2 Since f'(-4)<0 the function is decreasing in the interval (-oo,-3).

Choose a value for x that is greater than 0.

f'(0)=6 Since f'(0)>0 the function is increasing in the interval (-3, `oo).`

Because the function changed direction from decreasing to increasing a relative minimum will occur at x=-3. The relative minimum is the point (-3, 1).

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