Derivative (of p with respect to x) is right except for sign.
The problem is to maximize something. First thing you need to do is write an expression for what is to be maximized. Is it p or something else? Second step comes from calculus. Remember how to find the min or max of a function? You correctly imply that a derivative is involved, but what about the derivative?
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Optimization is the process of finding the best possible solution to a problem. In the context of differentiation, optimization involves using the derivative to find the maximum or minimum value of a function. This can help us find the most efficient solution to a problem.
Optimization problems can be found in many fields, including engineering, economics, biology, and physics. Some common examples include minimizing cost or maximizing profit in a business, finding the optimal route for a delivery truck, and maximizing crop yield in agriculture.
Local optimization involves finding the maximum or minimum value of a function within a specific interval. Global optimization, on the other hand, involves finding the maximum or minimum value of a function over its entire domain. This can be more challenging and often requires advanced techniques.
In order to find the optimal solution, you need to find the point where the derivative of the function is equal to zero. This point is called a critical point and can be either a maximum or a minimum. To determine which one it is, you can use the first or second derivative test.
While differentiation is a powerful tool for solving optimization problems, it does have some limitations. It may not always be possible to find an analytical solution, and in some cases, numerical methods may be needed. Additionally, optimization problems with multiple variables can be more complex and may require advanced techniques.