In my free response, I explain that I use a process approach to problem solving. My approach to calculus optimization problems is one idea of my process approach:
Process for Optimization Problems:
1. Determine any limitations that bound the problem.
2. Determine the function that we are interested in optimizing -- make sure it is in 1 variable. f(x)
3. Take the first derivative of the function => f ' (x).
4. Look for critical points where f '(x)=0 or f '(x)=infinity. Solve for appropriate x values (ones that make sense for the problem).
5. Take the second derivative f ''(x) to determine if the x's found in step 4 are max's or min's. Remember that f ''(x) gives concavity of the function f(x).
6. Plug in the critical point values of x into f ''(x). If f ''(x)>0 ==> min value (i.e. the curve is concave upward at x); if f ''(x)<0 ==> max value (i.e. the curve is concave downward at x).
I hope this helps all you calculus students out there. This is how I teach math at the high school level, through processes.