00:01
So the unit problem, let's call two numbers x and y.
00:05
And we know that the square of the first number plus the sum plus the second number, so x4 plus y, this should be equal to 54.
00:16
And we're trying to find these numbers such that their product, x y, so i'll write t equals x y, the product, is maximized.
00:28
So this is the equation we want to you've tacking this on them.
00:33
To do that we need to write it as a function of a single variable.
00:36
So let's say we want to write it as a function of x, we use this first equation up here and solve for y so that we can substitute.
00:45
So popping for y we get 54 minus x squared.
00:51
And talking that in down here, give x times 54 minus x squared.
00:58
And let's distribute that so we'll get 54 x minus x squared.
01:00
And let's distribute that so we'll get 54 x minus x cubed.
01:05
Right.
01:06
Next we need to find the critical number of this function.
01:09
So we begin by taking the derivative.
01:13
The p prime of x equals using the power rule gives us 54 minus 3x squared.
01:21
And then to find a critical number, those occur when the derivative is 0 or undefined.
01:28
In this case we only check when it's equal to 0.
01:33
So 54 minus 2x squared.
01:35
And then solving for x squared, we move the 54 on the other side so we get through x squared that equals 54 so x squared equal 18 giving us two solutions x equals plus or minus square of 18 which can be simplified to plus or minus 18 is 9 times 2 so this will be 3 times square root of 2 all right now one of these numbers will figure out which of these numbers is actually the maximum we can apply the number line test for the first derivative...