Video: Finding Two Numbers with the Maximum Product given Their Sum Using Differentiation

Find the two numbers whose sum is 96, and their product is as maximum as possible.


Video Transcript

Find the numbers whose sum is 96 and their product is as maximum as possible.

To start this problem off, what I’m going to say is that 𝑥 is one number. And therefore, 96 minus 𝑥 must be the other number. That’s because the sum of those numbers needs to equal to 96. And one way we can check that by just adding our values together. So if we get 96 minus 𝑥 plus 𝑥, well if we subtract 𝑥 and then add 𝑥 on, they’re going to cancel each other out. So we get the answer 96. So yes, we can say the sum is 96.

Okay, great, so now let’s move on to the next step. Well, if we look back at the question, we want to actually find the product of our numbers. We want to find out where it is actually as maximum as possible. So I’m gonna call our product 𝑃. And we can say that 𝑃 is equal to 𝑥 multiplied by 96 minus 𝑥. That’s because in product, we multiply them together. So therefore, we can say that 𝑃 is equal to 96𝑥 minus 𝑥 squared.

And it’s at this point, we can actually have a look at this sign here — the negative sign — because our question actually asks us to find where the product is as maximum possible. If we actually think about the shape of the function, if we had a U-shaped parabola, it would be positive 𝑥 squared. However, ours is negative 𝑥 squared. So that would be an inverted U-shaped parabola. So we know that actually our function is gonna have a maximum point. And a maximum point is where the slope is equal to zero or 𝑚 is equal to zero.

Okay, great, so in order to find out where that points going to be, now we’re going to differentiate our function. So if we differentiate, we get 96 minus two 𝑥. And that’s because actually we’ve differentiated 96𝑥, which just gives us 96. And then, negative 𝑥 squared gives us negative two 𝑥. Well, as we said before, to find the maximum or critical point, what we actually want to do is set our differential equal to zero. So we get zero is equal to 96 minus two 𝑥.

So now, we’re gonna solve for 𝑥. So if we add two 𝑥 to each side, we get two 𝑥 is equal to 96. And then, we divide by two to find one 𝑥. So we get 𝑥 is equal to 48. So therefore, at the critical point or the maximum, we can actually say that one of our numbers is equal to 48 because we said right at the beginning the 𝑥 was going to be one number.

And now to find the second number, what we need to do is actually substitute 𝑥 is equal to 48 into 96 minus 𝑥, which is going to be 96 minus 48 which will be equal to 48. So therefore, we can say the numbers whose sum is 96 and product is as maximum as possible are 48 and 48.

And just to recap how we did that, first of all, we actually found an expression for each of our numbers using the information that the sum was 96. Then, we formed an equation for the product, which we then differentiated because actually we wanted to find the point where the slope was going to be equal to zero because that’s gonna be our critical point and in this case the maximum point. We then actually solved with the slope function being equal to zero and we found our 𝑥-value. And then, finally, we used that 𝑥-value — so our critical point value — to find the other value which we got from our expression that we’ve made in the first step. And then, we finally arrived at our answer, which was the two values are 48 and 48.

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