Question

Find two numbers differing by 46 whose product is as small as possible.

Answer 1

Answer:

The numbers are 23 and -23.

Step-by-step explanation:

If the two numbers are x and y, we can say that the product is

$P=x\cdot y$

We need to eliminate a variable. To do that we use the fact that the difference between the two numbers has to be 46.

$x-y=46$

So we can say $y=x-46$ and plug that into the equation for P

$P=x\cdot (x -46)=x^2-46x$

From the information given we want the product P to be minimized. For this we find the derivative

$\frac{d}{dx}(x^2-46x) =\frac{d}{dx}\left(x^2\right)-\frac{d}{dx}\left(46x\right)=2x-46$

Next, we find the critical points, we set the above equation equal to zero and solve for x

$2x-46=0\\2x-46+46=0+46\\2x=46\\\frac{2x}{2}=\frac{46}{2}\\x=23$

Since x = 23 is the only critical number, we can conclude that there’s actually an absolute minimum at x = 23

Thus, $x = 23$ and $y = 23-46=-23$

Answer 2

The two numbers differing by 46 whose product is as small as possible are 23 and -23

Let the unknown numbers be x and y

If the numbers differs by 46, hence;

$x-y=46$ ....................1

Let the product of both numbers be P, hence;

$xy=P$ ........................2

From equation 1; x = 46 + y ................. 3

Substitute equation 3 into 2 to have:

$x(x-46)=P\\P = x^2-46x$

If the product is as small as possible, hence $\frac{dP}{dx} = 0$

$\frac{dP}{dx}=46 - 2x = 0\\46-2x = 0\\46 = 2x\\x = 23$

Recall that x - y = 46

23 - y = 46

y = 23 - 46

y = -23

Hence the two numbers differing by 46 whose product is as small as possible are 23 and -23

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