Final answer:
To find the two numbers, set up equations based on the given information. Find the minimum value of the product using the vertex of a quadratic function.
Step-by-step explanation:
To find the two numbers, let's call them x and y, we need to set up a system of equations based on the given information.
We know that the two numbers differ by 10, so we can write the equation x - y = 10.
The other equation involves the product of the larger number (x) and twice the smaller number (2y). We want to find the minimum value of this product, so we can write the equation xy as a function of x.
The equation is given by f(x) = x(2(x-10)) = 2x(x-10).
To find the minimum value of f(x), we can find the vertex of the quadratic function.
The x-coordinate of the vertex is given by -b/2a, where a = 2, b = -40, and c = 0 in our case. Plugging in the values, we get x = -(-40)/(2*2) = 10.
Substituting x = 10 into the equation x - y = 10, we get 10 - y = 10, which gives y = 0.
Therefore, the two numbers are x = 10 and y = 0.