Solution assignment 02 Optimizing f(x)

Verify whether the graph of the function has extremes:

$y=x^3+x^2+10x+1$

and if so, whether it is a maximum of a minimum.

We find candidates for a maximum or minimum by solving the equation:

$y'=3x^2+2x+10=0$

This quadratic equation has no solutions because the discriminant:

$D=1-4\cdot1\cdot10<0$

The graph has no (local) extremes nor a horizontal inflection point.