Can a parabola both have a maximum and a minimum why or why not

No

Step-by-step explanation:

The max or min of a parabola is determined by a in the following general equation.

y = ax^2 + bx + c

a cannot equal 0. Otherwise you get a line from bx + c = y

if a<0 then you get a maximum

if a>0 then you get a minimum.

The red is a>0

The blue is a<0