Step 1: Locating the by-product of the feature, we have:

$ latex f( x)= x ^ 4 +4 x ^ 3 +1$

$ latex f'( x)= 4x ^ 3 +12 x ^ 2$

Step 2: Developing a formula with the by-product as well as addressing for x, we locate the fixed factors:

$ latex 4x ^ 3 +12 x ^ 2= 0$

$ latex 4x ^ 2( x +3)= 0$

$ latex x _ {1} =0 ~ ~$ or $latex ~ ~ x _ {2} =-3$

Step 3: The y-coordinates of the fixed factors are:

$ latex y _ {1} =( 0 )^ 4 +4( 0 )^ 3 +1$

$ latex y _ {1} =1$

$ latex y _ {2} =( -3 )^ 4 +4( -3 )^ 3 +1$

$ latex y _ {2} =-26$

Step 4: We utilize the 2nd by-product to identify the nature of the fixed factors:

$ latex f ^ {prime prime} (x)= 12x ^ 2 +24 x$

When $latex x= 0$, we have $latex f ^ {prime prime} (x)= 0$ as well as when $latex x= -3$, we have $latex f ^ {prime prime} (x)> > 0$.

The factor $latex x= 0$ might be a transforming factor. By utilizing the worths $latex x= 0.1$ as well as $latex x= -0.1$ in the initial by-product, we obtain inclines with the exact same indication, so the factor is undoubtedly an inflection factor.

The inflection factor collaborates are (0, 1) as well as the minimal factor collaborates are (-3, -26).



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