find two positive numbers whose sum is 32 and the sum of their square roots is maximum. (OMIT DOMAIN)

Answer 1

This is a pretty standard optimization problem in a single variable calculus class. We aim to find two numbers, which we'll call x and y, that satisfy the given conditions. We do this by taking the following steps:

(1) Setting up the problem.

(2) Getting a single variable function that we'll call f(x).

(3) Finding the derivative f'(x) and it's critical points.

Suppose that x + y = 32. We aim to optimize the expression √x + √y. The problem is that this is in terms of two variables. We'd really like it to be in terms of one variable; say x.

Notice that we can solve for y: y = 32 - x. This allows us to work with √x + √(32-x) instead of √x + √y. So define

f(x) = √x + √(32 - x)

We now have a one variable function which we can perform calculus on. Let's find the derivative of f(x). To make our lives easier, let's use rational notation instead of radical notation.

f(x) = (x)^1/2 + (32-x)^1/2

⇒ f ' (x) = 1/2 * [ ( 1 / √x ) - ( 1 / √(32 - x) ) ]

Slightly annoying function, but we have something we can optimize. Let's find this functions critical points. We set f'(x) = 0 and solve for 6.

Using algebra, we find that x = 16 is our only solution. Now, plugging 15 and 17 into f'(x) gives us the following:

f'(15) =  0.00783163 > 0

f'(17) =  -0.00783163 < 0

This tells us that x = 16 is a local maximum for f(x). Therefore, we can conclude that x = 16 and y = 32 - 16 = 16 and we're done!

Answer 2

The two numbers are x and 32 - x

f(x) = x(32 - x)

f(x) = 32x - x²

Determine f'(x), solve f'(x) = 0.