Suppose you want to find two positive numbers such that the sum of the first and twice the second is 64 and whose product is a maximum.
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To find two positive numbers such that the sum of the first and twice the second is 64 and whose product is a maximum, you can use algebraic optimization techniques.
Let's assume the first positive number is "x" and the second positive number is "y".
According to the given condition, the sum of the first number and twice the second number is 64. In equation form, this can be written as:
x + 2y = 64
To find the maximum product of x and y, we need to use a technique called "optimization". This involves finding a relation between the variables, substituting it into a function that represents the problem, and maximizing or minimizing that function subject to any given constraints.
Because we want to maximize the product xy, we can express xy as a function of one variable using substitution. Using the equation x + 2y = 64, we rearrange it to solve for x:
x = 64 - 2y
Now, we can substitute this value of x in terms of y into the function xy:
f(y) = x * y = (64 - 2y) * y = 64y - 2y^2
To find the maximum value of f(y), we find the critical points by taking the derivative of f(y) with respect to y and setting it equal to zero:
f'(y) = 64 - 4y = 0
Solving this equation yields y = 16.
Now, we substitute this value of y back into the equation x = 64 - 2y:
x = 64 - 2 * 16 = 32
So, the two positive numbers that maximize the product are x = 32 and y = 16.