btw i'm using periods for number readability instead of commas since i use a lot of commas, so watch out for that

if i use the example of our unit 4 test, put pretty basically:

the problem asks you to **maximize** revenue sold from tickets. starting from $12, every dollar you take off the ticket price will have attendance be raised by 1000, with $12 having an average attendance of 11.000. have ticket price equal 12-x, and the average attendance equal 11.000 + 1.000x.
since you are looking to maximize *revenue*, you want to multiply them together in this case. if your optimization problem only has a single equation and it's directly related, don't bother looking for a second problem to multiply.
when you have your full equation (-1.000x^2 + 1.000x + 132.000), get its derivative and find the derivative's critical number(s). in case you don't remember, to find the critical number solve the equation to zero (or undefined). this gives us 1/2 as our critical number.

once you have a critical number, you have to prove it's either concave up or concave down. you can do this one of two ways:
1) the derivative of the original (-1.000x^2 + 1.000x + 132.000 if you don't want to look back up) can be derived again to check for concavity. it comes out to -2.000, which is negative. since the second derivative measures concavity, this means the graph is always negative. this is good, since it means its critical point is also its absolute maximum. (not every optimization problem will give a flat double derivative. if there's still an x left over, plug in your critical number)
2) keeping with the first derivative, plug in any two numbers higher or lower than your critical number. if you do this, do easy numbers. if i do 0 and 1 for example, the answers come out to be 1000 and -1000 respectively. since it goes from positive to negative, it has negative concavity (this has to be memorized). if it went negative to positive, it has positive concavity.

either way, you know it's negative concavity. this means you can be confident your critical number (as x on the original equation) is actually the maximum of the equation. if you plug the answer in, you have your average maximum revenue of $13.250.