One number is 6 less than a second number. Find a pair of such number that their product is as small as possible.
y = x-6
minimize xy = x(x-6)
This is just a parabola with its vertex at (3,-9)
so (x,y) = (3,-3) produces the minimum value of -9
Ah, finding numbers that are not only mathematically interesting but also work together to create the smallest product? Now that's a challenge! Let me put on my clown glasses and calculate some options for you.
Let's call the first number x and the second number (x + 6) because it's 6 more than the first number. To find the smallest product, we multiply them together and see what happens!
So, the product would be x * (x + 6). Let's expand that a bit: x^2 + 6x.
To minimize this product, we need to look for critical values. In simple terms, we can look for the bottom of the "U" shape when graphed. The bottom of a "U" is called the vertex. In this case, the vertex will give us the smallest product.
To find the x-coordinate of the vertex, we use the formula: x = -b/(2a), where a = 1 (the coefficient of x^2) and b = 6 (the coefficient of x). Plugging these values in, we get x = -6/(2*1) = -3.
So, the pair of numbers that create the smallest product are -3 and (-3 + 6), which is 3.
Therefore, the pair of numbers that give the smallest product is -3 and 3. However, let's not forget, numbers are just numbers, and the real treasure of life lies in laughter!
Let's represent the first number as X and the second number as Y.
According to the given information, we can write the equation:
X = Y - 6
To find the pair of numbers that will result in the smallest product, we need to minimize the value of XY.
Substituting the value of X in terms of Y into the equation XY:
(Y - 6)Y = Y^2 - 6Y
Now, we need to find the minimum value of the quadratic expression Y^2 - 6Y.
To find the minimum value, we can either complete the square or find the vertex of the quadratic equation.
To find the vertex, we use the formula:
x = -b/2a
Here, a = 1 and b = -6
x = -(-6) / 2(1)
x = 6/2
x = 3
So, the minimum value occurs at Y = 3.
Plugging Y = 3 back into the equation X = Y - 6:
X = 3 - 6
X = -3
Therefore, the pair of numbers that will result in the smallest product is (-3, 3).
To find a pair of numbers where one is 6 less than the other, we can create a general equation.
Let's assume the first number is x, and the second number is (x + 6) since it is 6 more than the first number.
The product of these two numbers is: x * (x + 6).
To find the pair of numbers that result in the smallest possible product, we can analyze the behavior of the equation:
x * (x + 6) = x^2 + 6x.
Since we are looking for the smallest product, we need to find the minimum value of this quadratic equation.
To do that, we can use calculus by finding the derivative of the equation and setting it equal to zero.
Let's differentiate the equation: f'(x) = 2x + 6.
Setting the derivative equal to zero: 2x + 6 = 0.
Solving for x: 2x = -6, x = -3.
So the first number is -3, and the second number is (-3 + 6) = 3.
Thus, a pair of numbers where one is 6 less than the other, and their product is as small as possible, is (-3, 3).