Answer:
2500
Step-by-step explanation:
We have to find the largest product of two numbers whose sum is 100.
Let the two numbers be x and y.
Thus, we can write x+y=100
We can calculate the value of y as:
y = 100 - x
The product of these number can be written as: (x)(y) = (x)(100-x) = 100x - x²
Let f(x) = 100x - x²
Now, the first derivative of this function with respect to x is
[tex]\frac{df(x)}{dx}[/tex] = 100-2x
Equating [tex]\frac{df(x)}{dx}[/tex] = 0, we get,
100-2x = 0
⇒ x = 50
Now, we find the second derivative of the the function f(x) with respect to x
[tex]\frac{d^2f(x)}{dx^2}[/tex] = -2
Since, [tex]\frac{d^2f(x)}{dx^2}[/tex] < 0, then by double derivative test the function have a local maxima at x = 50
This, x = 50 and y = 100-50 =50
Largest product = (50)(50) = 2500
