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

Video Transcript

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

In this question, we are asked to find the two numbers, which we will call 𝑎 and 𝑏, whose sum is equal to 96 and whose product is as large as possible. We could try to solve this problem by substituting in values and using trial and improvement. However, in this case, we’ll use a more formal method involving differentiation. We will begin by letting the first number 𝑎 equal 𝑥. Then, since 𝑎 plus 𝑏 equals 96, 𝑏 is equal to 96 minus 𝑥. We can check this by adding 𝑥 and 96 minus 𝑥, which does indeed give us a sum of 96. Next, let’s consider the product of the two numbers. If we let this product be 𝑃, then 𝑃 is equal to 𝑥 multiplied by 96 minus 𝑥. Distributing the parentheses or expanding the brackets, this becomes 96𝑥 minus 𝑥 squared.

At this point, it’s worth noting the sign in front of the quadratic term. The question asks us to find the values where the product is as large as possible. This is equivalent to the maximum value. And we recall that if we have a quadratic function as in this case and the 𝑥 squared term is negative, we will have an n-shaped parabola. This will have a maximum point. And we know that the slope or gradient 𝑚 at this point will equal zero. We know that we can find the gradient of any function by differentiating. Using the power rule of differentiation, if 𝑃 is equal to 96𝑥 minus 𝑥 squared, then 𝑃 prime or d𝑃 by d𝑥 is equal to 96 minus two 𝑥.

We can then find the maximum or critical point by setting this equal to zero. If zero is equal to 96 minus two 𝑥, then adding two 𝑥 to both sides, we have two 𝑥 is equal to 96. And dividing through by two, 𝑥 is equal to 48. This means that at the critical point or the maximum, 𝑥 is equal to 48. We recall that we let the first of our two numbers 𝑎 be equal to 𝑥. Therefore, 𝑎 is equal to 48. And since the second number 𝑏 is equal to 96 minus 𝑥, this is equal to 96 minus 48, which is also equal to 48. We can therefore conclude that the two numbers whose sum is 96 and their product is as large as possible are 48 and 48.

We were able to do this by finding an expression for the product, differentiating this, and then setting the derivative equal to zero, as this gave us the point where the slope or gradient was equal to zero and in turn the maximum point.