`f(x) = x^4 - 32x + 4` 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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Given:  `f(x)=x^4-32x+4`

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

`f'(x)=4x^3-32=0`

`4x^3=32`

`x^3=8`

`x=2`

The critical value is x=2.

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.

f'(0)=-32 Since f'(0)<0 the function is decreasing on the interval (-`oo,2).`

Choose a value for x that is greater than 2.

f'(3)=76 Since f'(3)>0 the function is increasing on the interval (2, `oo).`

Since the function changed directions from decreasing to increasing a relative minimum exists. The relative minimum occurs at the point (2, -44).