obteber dos números tales que la suma de uno de ellos con el doble del cuadrado del otro sea igual a 7776, y el producto de ambos sea máximo

Answer: the numbers are 5,111.3 and 35.74

Step-by-step explanation:

I will answer this in English.

We have the numbers X and Y, ad we have that:

X + 2*Y^2 = 7776

X = 7776 - 2*Y^2

Now we also want that the product of X and Y is a maximum.

The equation of maximum product is

P (Y) = X*Y = (7776 - 2*Y^2) * Y

= 7776*Y - 2*Y^3

Now, to find the maximum we must derivate and find the value where P' (Y) is equal to 0

P' (Y) = 7776 - 6*Y^2 = 0

7776 = 6*Y^2

Y = √ (7776/6) = 35.74

Now, we replace this value in the equation for X and we can find X:

X = 7776 - 2*Y^2 = X = 7776 - 2 * (35.74) ^2 = 5,111.3