Find the absolute minimum value on (0,infinity) for f(x)=(x+8)(x-4)^2

It is a continuous function whose derivative exists.  Therefore, extreme values are either on the edge of the interval [0 , infinity], or at the zero slope points where the derivative is zero.  So calculate the value of the function at x = 0;  then look at the value of the function when x increases to infinity (larger and larger values).  Looks like the larger x is, the larger the function, so it tends to + infinity as well.  Thankfully, we are looking for an absolute minimum, therefore the infinity value is out. Then calculate the derivative, and find the x values for which it is zero.  Then calculate the value of the function for those zero slope x values, and compare to the value of the function at the edge, x = 0.  Finally, pick the x value for which the function value is lowest.  That is your absolute minimum value.

Hope this helps!! (If not I'm sorry!)