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among all pairs of numbers whose difference is 20, find a pair whose product is as small as possible. what is the minimum product?

The pair of numbers whose difference is 20 and whose product is as small as possible is ____.

Respuesta :

mathstudent55

Answer:

-100

-10, 10

Step-by-step explanation:

Let the smaller number be x.

Then the larger number is x + 20.

The product is x(x + 20).

Now you can write the function

y = x(x + 20)

y = x^2 + 20x

Take the first derivative of y with respect to x.

y' = 2x + 20

Set the first derivative equal to zero to find the x value for the minimum value of the function.

2x + 20 = 0

2x = -20

x = -10

The minimum value of the function occurs at x = -10. -10 is one of the two numbers.

y = x^2 + 20x

For x = -10,

y = (-10)^2 + 20(-10)

y = 100 - 200

y = -100

The minimum value of the product is -100.

x = -10

x + 20 = -10 + 20 = 10

The numbers are -10 and 10.

If you have not learned derivatives yet, then plot the function

y = x^2 + 20x

Now look at the graph and find the minimum y value and the x value at which it occurs.

The minimum y value is -100. That is the minimum product you are looking for.

The x value of the minimum function value is x = -10.

Then x + 20 = -10 + 20 = 10.

The numbers are -10 and 10.

facundo3141592

We want to find two numbers such that their difference is 20 and whose product is minimized.

The pair of numbers is 10 and -10.

Let's define our two numbers as A and B.

Such that A < B.

We want the difference to be equal to 20, then we have:

B - A  = 20.

The product between these two numbers is written as:

P = A*B

From the first equation, we can write:

B = 20 + A

Now we can replace that on the product equation to get:

P = A*(20 + A) = 20*A + A^2

Now we want to minimize this, notice that this is a quadratic polynomial, thus the minimum is at the vertex.

For a general quadratic polynomial.

y = a*x^2 + b*x + c

The vertex is at:

x = -b/2a

So in our case, P = 20*A + A^2

The vertex is at:

A = -20/(2*1) = -10

Then we have A = -10

To find the value of B we use the first equation:

B - A = 20

B = 20 + A = 20 + (-10) = 10

Then the pair of numbers is A = -10 and B = 10, and the product is:

P = 10*-10 = -100.

If you want to learn more, you can read:

https://brainly.com/question/2866380

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