Application of Maxima and Minima

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As an example, the area of a rectangular lot, expressed in terms of its length and width, may also be expressed in terms of the cost of fencing. Thus the area can be expressed as A = f(x). The common task here is to find the value of x that will give a maximum value of A. To find this value, we set dA/dx = 0.
 

Steps in Solving Maxima and Minima Problems

  1. Identify the constant, say cost of fencing.
  2. Identify the variable to be maximized or minimized, say area A.
  3. Express this variable in terms of the other relevant variable(s), say A = f(x, y).
  4. If the function shall consist of more than one variable, expressed it in terms of one variable (if possible and practical) using the conditions in the problem, say A = f(x).
  5. Differentiate and equate to zero, dA/dx = 0.

Discussion

Okay, so I got really confused at part 1 of the problems.

There are some problems that differentiate likes this:
Problem 8
Let x and y the numbers
x+y=a
1+y′=0
y=−1

and there are some that do this:
Problem 6
Let x and y = the numbers
x+y=a
x=a−y

how do I know which one to use and when?

I'm also confused on that, since I'm using a lot of book reference they don't use derivatives for the first equation. They only equate it to zero (derive) on the equation to be maximized/ minimized.

I think it is about implicit and explicit type o differentiation. The first is differentiation under several variables, the other one is differentiation under single variable. When to use which, depends on the equation. Implicit differentiation will simplify complex process in explicit. There are times that it is is the other way around.

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