### 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 π and π, 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, weβll use a
more formal method involving differentiation. We will begin by letting the first
number π equal π₯. Then, since π plus π equals 96,
π is equal to 96 minus π₯. We can check this by adding π₯ and
96 minus π₯, which does indeed give us a sum of 96. Next, letβs consider the product of
the two numbers. If we let this product be π, then
π is equal to π₯ multiplied by 96 minus π₯. Distributing the parentheses or
expanding the brackets, this becomes 96π₯ minus π₯ squared.

At this point, itβs 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 π₯ 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 π 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 π is equal to 96π₯ minus π₯ squared, then π prime or dπ by
dπ₯ is equal to 96 minus two π₯.

We can then find the maximum or
critical point by setting this equal to zero. If zero is equal to 96 minus two
π₯, then adding two π₯ to both sides, we have two π₯ is equal to 96. And dividing through by two, π₯ is
equal to 48. This means that at the critical
point or the maximum, π₯ is equal to 48. We recall that we let the first of
our two numbers π be equal to π₯. Therefore, π is equal to 48. And since the second number π is
equal to 96 minus π₯, 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.