Find two numbers which have a difference of 7, and a product that is a minimum. Thank you! :)
Ah, the quest for minimum product and maximum entertainment! Let's dive into the mathematical circus, shall we?
Picture this: You're lost in the enchanted forest counting on your fingers when suddenly you stumble upon two mischievous numbers, let's call them "N" and "M". They're looking for a perfect balance of difference and product to create some mathematical magic.
Now, let's get down to clown business. We know that the difference between N and M is 7. So, we can write it as an equation: N - M = 7.
To find the minimum product, we need to make sure our numbers are as close together as possible. So, let's take a guess! What if we assign N as a slightly bigger number and M as slightly smaller?
Let's make N = 4 and M = -3. Now, let's check if these numbers fit our magical equation: 4 - (-3) = 7. Success! The difference is indeed 7.
Now, let's move to the grand finale, the product! To find it, we multiply N and M: 4 * (-3) = -12.
So, we have our magical pair! N = 4 and M = -3, with a difference of 7 and a product of -12. Voila! Isn't math amazing when it's combined with a little bit of clownery? Enjoy your numerical journey!
To find two numbers with a difference of 7 and a minimum product, we can start by letting one of the numbers be x. The other number will then be (x + 7) since their difference is 7.
To find the minimum product, we can take the derivative of their product with respect to x and set it equal to zero. This will give us the minimum value of the product.
Let's proceed with finding the minimum product step-by-step:
1. Let one of the numbers be x.
2. The other number will be (x + 7).
3. The product of these two numbers is given by: P(x) = x(x + 7).
4. Expand the product: P(x) = x^2 + 7x.
5. Take the derivative of P(x) with respect to x: P'(x) = 2x + 7.
6. Set P'(x) equal to 0 and solve for x:
2x + 7 = 0
2x = -7
x = -7/2
7. Calculate the other number by adding 7 to x:
Other number = (-7/2) + 7 = 7/2
8. Therefore, the two numbers with a difference of 7 and a minimum product are -7/2 and 7/2.
Thus, the two numbers are -7/2 and 7/2.
To find two numbers with a difference of 7 and a minimum product, let's call the larger number x and the smaller number y. Since their difference is 7, we can write the equation x - y = 7.
To find the minimum product, we need to minimize the value of xy. We can rewrite this equation as: x = y + 7.
Now, we substitute y + 7 for x in the equation xy to get our equation for the product: P = (y + 7)y.
To minimize the product, we need to find the critical points of this equation. To do this, we take the derivative of P with respect to y and set it equal to zero:
dP/dy = 2y + 7 = 0.
Solving this equation, we find that y = -7/2.
Substituting y back into the equation x = y + 7, we find:
x = (-7/2) + 7 = 17/2.
Therefore, the two numbers with a difference of 7 and a minimum product are y = -7/2 and x = 17/2.