To solve optimization problems in calculus, follow these simple steps:

1. Understand the Problem:

   • Read the problem carefully.
   • Figure out which function you need to make as big (maximize) or as small (minimize) as possible.
   • Check for any limits or conditions that might change how you can solve it.

2. Identify Variables:

   • Name the important things (variables) in the problem using letters like $x$ or $y$.
   • If there are limits, try to write your function using just one variable if you can.

3. Formulate the Function:

   • Write down the main function you're trying to work with, called $f(x)$.
   • Use any limits to help you express $f(x)$ the right way.

4. Differentiate:

   • Find the derivative of your function, called $f'(x)$. This helps you see how the function changes.
   • Set the derivative equal to zero: $f'(x) = 0$. This step finds where the function might reach its highest or lowest points.

5. Find Critical Points:

   • Solve the equation $f'(x) = 0$. This will help you find points that could be the maximum or minimum values.
   • Also, check the edges of your range to see how they affect the function.

6. Second Derivative Test:

   • Look at the second derivative, $f''(x)$. This tells you more about your critical points.
     • If $f''(x) > 0$, you have a local minimum (the lowest point).
     • If $f''(x) < 0$, you have a local maximum (the highest point).

7. Evaluate and Conclude:

   • Put the critical points back into the original function to find the highest or lowest values.
   • Clearly show your results by stating what the best solution is and any important conditions to consider.