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Viktor [21]
3 years ago
9

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

t?
The pair of numbers whose difference is 20 and whose product is as small as possible is ____.
Mathematics
1 answer:
Elza [17]3 years ago
4 0

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.

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