The optimization problem is considered finished when you have found the maximum or minimum value of the function, and you have also found the corresponding values for the variables that give that maximum or minimum value. In other words, you have found the optimal solution.
Calculus is used in optimization problems to find the maximum or minimum value of a function by finding the critical points, which are points where the derivative of the function is equal to zero. The first derivative test is then used to determine if the critical point is a maximum or minimum.
To check the solution for an optimization problem, you can plug the values for the variables into the original function and see if it gives the maximum or minimum value. You can also take the second derivative of the function at the critical point to confirm if it is a maximum or minimum.
Yes, optimization problems can have multiple solutions. This can occur when there are multiple critical points or when the function has a flat section where the derivative is equal to zero. In these cases, it is important to check each solution to determine which one is the global maximum or minimum.
Some common mistakes to avoid when solving optimization problems include not checking all critical points, not considering the domain of the function, and not correctly setting up the equations or inequalities. It is also important to carefully interpret the results and make sure they make sense in the context of the problem.