[music] Here's kind of an old timey puzzle about numbers.
Suppose you've got 2 numbers which add up to 24.
You've got 2 numbers that sum to 24. How large can their product be?
When you take those 2 numbers that add up to 24, how big of a number can you get
when you multiply them together? This is an optimization problem so let's
draw a picture. Okay, maybe I can't draw a picture in this
problem but at least I can label things. So, instead of just saying two numbers
that sum to 24, let's give them the name. Two numbers, x and y, which sum to, to 24.
And instead of just saying how large can their product be, I can say, how large
can, you know, x times y be? So, what's the goal?
Well, my goal here is really just to maximize the product, so maximize x times
y. And there's a constraint.
The constraint is that x and y have to add up to 24.
So, I'll write that as x plus y equals 24. Rats.
X,y isn't a function of a single variable. I need to rewrite it so it's a function of
x alone. Now, since x plus y is 24, I could rewrite
this as y equals 24 minus x. And consequently, x times y, which is the
thing I'm trying to maximize, I could write that as a function of a single
variable, x times, and instead of y 24 minus x.
So, this is the quantity now that I want to maximize.
Now, I can apply Calculus. Let's differentiate and then find the
critical points. So, here's this function of a single
variable that I'm trying to maximize. My function is x times 24 minus x, that
should be y or this is x times y, this thing I'm trying to maximize.
But if I expand this out, I get 24x minus x squared.
Then, I can easily differentiate this. The derivative of this function is, what's
the derivative of 24x? It's 24.
What's the derivative minus x squared? It's minus 2x.
I'm trying to find critical points, right? Critical points are where the derivative
doesn't exist or where the derivative is equal to 0.
Now, this function's just a polynomial, so it's differentiable everywhere.
So, I don't have to worry about the functions derivative not existing
somewhere. But I do have to find places where the
derivative is equal to zero. So, when is this thing equal to zero?
Well, that's the same as asking, when 24 is 2x?
That's exactly when x is 12. So, here is my critical point.
What sort of point is that? Well, let's think about the s, i, g, n,
the sign of the first derivative. The derivative is positive if x is less
than 12, and the derivative is negative if x is bigger than 12.
What does that mean? Well, that means the function is
increasing for x values less than 12. And the function is decreasing for x
values bigger than 12. So, the function goes up, gets to 12, and
starts going down. What kind of point does that make 12?
Well, that must be a maximum. So, I'll write f of 12 is my maximum
value. So, what do we conclude?
So, here's our conclusion. A 144 is the maximum product of two
numbers which sum to 24. Did we really need the awesome power of
Calculus to solve this problem? No.
For example, you could have used a technique like this.
There's the so-called arithmetic geometric mean inequality.
The AM-GM inequality. And what it tells you is the following.
That for numbers a and b, the arithmetic mean, just a fancy word for the average of
a and b, is bigger than or equal to the geometric mean, which is the square root
of the product of a and b. And this inequality becomes an equality,
if and only if a and b are the same. Now, how can you use something like this?
Well, in our specific case, what do we know?
I know that x plus y is equal to 24 and if these two numbers add to 24, then their
average x plus y over 2 must be equal to 12.
But if this is their average by the AM-GM inequality, this must be bigger than or
equal to the square root of x times y, the geometric mean of x and y.
Now, how does this help us? Well, what if I square both sides, right?
Then I know that 144 is bigger than or equal to x times y.
And that's exactly what I'm trying to figure out.
I'm trying to figure out how big can the product of x and y be, if I know what the
sum is. And what this is telling me, is that the
product can be no bigger than 144, with equality when x and y are both equal.
And that happens when they are both equal to 12.
That you don't actually need Calculus to solve many of the problems that are
assigned in Calculus courses, is perhaps one of the best kept secrets of Calculus
instructors. A result like this AM-GM inequality,
actually solves a ton of optimization problems.
You often don't have to resort to taking derivatives, if you can just apply a
result like this. Calculus is a powerful tool but there are
other ways to attack these problems. Don't be afraid to make use of other
methods.