`f(x) = -2x^2 + 4x + 3` 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 19 - 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)=-2x^2+4x+3`

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)=-4x+4=0`

`-4x=-4`

`x=1` 

The critical value for the first derivative is x=1.

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

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

Choose a value for x that is less than 1.

f'(0)=4 Since f'(0)>0 the function increases in the interval (-oo, 1).

Choose a value for x that is greater than 1.

f'(2)=-4 Since f'(2)<0 the function decreases in the interval (1, `oo).`

Because the function changed directions from increasing to decreasing a relative maximum will exist.  The relative maximum is the point (1, 5).

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