Critical points occur when [tex]f'(x)=0[/tex], which happens for [tex]x=-\dfrac53[/tex] and [tex]x=1[/tex].
Check the sign of the second derivative at each critical point to determine the function's concavity at that point. If it's concave ([tex]f''(x)<0[/tex]), then a maximum occurs; if it's convex ([tex]f''(x)>0[/tex]), then a minimum occurs.
You have
[tex]f''(x)=6x+2[/tex]
and so
[tex]f''\left(-\dfrac53\right)=-8<0\implies f(x)\text{ is concave at }x=-\dfrac53[/tex] [tex]f''(1)=8>0\implies f(x)\text{ is convex at }x=1[/tex]
This means a maximum of [tex]f\left(-\dfrac53\right)=\dfrac{121}{27}[/tex] and a minimum of [tex]f(1)=-5[/tex].
