Question

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

Answer 1

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

$\frac{df(x)}{dx}$ = 100-2x

Equating $\frac{df(x)}{dx}$ = 0, we get,

100-2x = 0

⇒ x = 50

Now, we find the second derivative of the the function f(x) with respect to x

$\frac{d^2f(x)}{dx^2}$ = -2

Since, $\frac{d^2f(x)}{dx^2}$ < 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