(b) What's the largest product possible from two numbers adding up to 100?

Accepted Solution

Answer:2500Step-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=100We can calculate the value of y as:y = 100 - xThe product of these number can be written as: (x)(y) = (x)(100-x) = 100x - x²Let f(x) = 100x - x²Now, the first derivative of this function with respect to x is[tex]\frac{df(x)}{dx}[/tex] = 100-2xEquating [tex]\frac{df(x)}{dx}[/tex] = 0, we get,100-2x = 0⇒ x = 50Now, we find the second derivative of the the function f(x) with respect to x[tex]\frac{d^2f(x)}{dx^2}[/tex] = -2Since, [tex]\frac{d^2f(x)}{dx^2}[/tex] < 0, then by double derivative test the function have a local maxima at x = 50This, x = 50 and y = 100-50 =50Largest product = (50)(50) = 2500