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).