`f(x) = -3x^2 - 4x - 2` 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 20 - 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)=-3x^2-4x-2`

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)=-6x-4=0`

`-6x=4`

`x=4/-6`

`x=-2/3`

The critical value for the first derivative is x=-2/3. 

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 for x that is less than -2/3.

f'(-1)=2 Since f'(-1)>0 the graph of the function is increasing in the interval

(-oo,-2/3).

Choose a value for x that is greater than -2/3.

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

(-2/3, `oo).`

Because the direction of the function changed from increasing to decreasing a relative maximum will exist at x=-2/3. The relative maximum is the point (-2/3, -2/3).

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