Question

b. Find a pair of numbers that have a sum of 50 and will produce the largest possible product. Example: +_ = 50 (sum) so _* _ = _ (maximum area) and (enter answers from the sum)

Answer 1

A pair of numbers that have a sum of 50

Let the number is x, so the other number is 50 - x

Let f(x) be the largest product so:

`f(x)=\text{ x(50-x)}`

Simplify the expression :

`\begin{gathered} f(x)=\text{ x(50-x)} \\ f(x)=50x-x^2 \end{gathered}`

Diffrentiate with respect to x

`\begin{gathered} f(x)=\text{ x(50-x)} \\ f(x)=50x-x^2 \\ \text{ Diffrentiate with respect to x} \\ f^(\prime)(x)=50-2x \\ \text{Apply derivative equal to zero:} \\ 50-2x=0 \\ 50=2x \\ x=25 \end{gathered}`

Now for to check for the f(x) is maximum for x = 25

Calculate the second derivative and put x = 25 is the f(x) is negative then the multiplication f(x) is maximum

`\begin{gathered} f^(\prime)(x)=50-2x \\ \text{ Differentiate with respect to x} \\ f^(\prime)^(\prime)(x)=0-2 \\ \text{ Substitute x = 25} \\ f^(\doubleprime)(25)=-2 \\ f^(\doubleprime)(25)<0 \\ \text{Thus the function f(x) is maximum for x = 25} \end{gathered}`

Thus, the first number is 25

Second number is : 50 -x = 50-25 = 25

Numbers are 25, 25

Answer : 25 + 25 =50 (sum)

25 * 25 = 625 (maximum possible product)