divide 50 into two parts such that the product will be a maximum
one part --- x
the other part ---- 50-x
product = y = x(50-x)
y = -x^2 + 50x
this is a downwards opening parabola, so it has a maximum
Use your favourite method to find that vertex and the mystery
will be revealed to you.
To divide 50 into two parts such that the product is a maximum, we can solve this problem using optimization techniques. Let's call the first part of the division x and the second part 50 - x.
To find the maximum product, we need to maximize the function f(x) = x(50 - x). We can do this by finding the critical points of the function, which occur when the derivative of the function is equal to zero.
Let's differentiate f(x) with respect to x:
f'(x) = 50 - 2x
Setting f'(x) = 0, we have:
50 - 2x = 0
Solving for x, we get:
2x = 50
x = 50/2
x = 25
So, the critical point is x = 25.
To determine whether this point is a maximum or minimum, we can check the second derivative of f(x).
Let's differentiate f'(x) with respect to x:
f''(x) = -2
Since the second derivative is negative, the critical point x = 25 corresponds to a maximum value for the function f(x).
Therefore, to divide 50 into two parts such that the product is a maximum, the two parts should be 25 and 50 - 25, or simply 25 and 25.