How do you solve this problem using derivatives find two real numbers x and y so that x-y equals 100 and their product is minimized?

Answer 1

At first glance, this would appear to be a multivariable problem. But in actuality, with the two equations we have to work with:

f(x,y) = xy

x - y = 100

We can rearrange the second one to find what y is equal to in terms of x:

y = 100 - x

(We can also solve x in terms of y, but solving y in terms of x is conventional).

then we can plug this in for y into f(x,y) to make it simply a function of x:

f(x) = x(100 - x) = 100x - x2

We then take the derivative of this, setting f'(x) equal to 0:

f'(x) = 0 = 100 - 2x

2x = 100

x = 100/2 = 50

This is the value of x that minimizes the product of x and y when x - y = 100. We then solve for y by plugging in the x-value into this equation:

50 - y = 100

-y = 100 - 50

y = -50

By plugging in values for which x > 50 and x < 50 into the equation f(x) = 100x - x2, we find that f(x) is indeed greater on both the intervals where x < 50 and where x > 50, proving that x = 50 is indeed an absolute minimum.