A common mistake is failing to correctly identify the objective function (the quantity to be maximized or minimized) and the constraint equation (the equation that limits the possible values of the variables).
Double-check your differentiation using the chain rule, product rule, and quotient rule as needed. Consider using a computer algebra system (CAS) like Desmos or Wolfram Alpha to verify your derivatives.
In closed interval optimization problems, the absolute maximum or minimum may occur at the endpoints of the interval, not just at critical points. Failing to check endpoints can lead to an incorrect solution.
If a critical point is outside the feasible domain (i.e., it doesnt satisfy the constraint or any other restrictions), discard it. It cannot be a solution to the optimization problem.
Use the constraint equation to eliminate one variable, expressing the objective function in terms of a single variable. Then, proceed with finding critical points and checking endpoints.
After finding the optimal value, make sure to answer the original question asked in the problem. State the optimal values of all relevant variables, including units, and explain what they mean in the context of the problem.