Question

What two nonnegative real numbers with a sum of 30 have the largest possible product? Let x be one of the numbers and let P be the product of the two numbers. Write the objective function in terms of x. P= (Type an expression.) The interval of interest of the objective function is (Simplify your answer. Type your answer in interval notation) The numbers that have a sum of 30 and have the largest possible product are (Use a comma to separate answers as needed.) Enter your answer in each of the answer boxes.

Answer 1

Solution:-

Assume be the one number

T

and be another number

Sum of the numoer =

30

C +y = 30

$y=30-x$ --------------------------------(1)

The product of both the nuber is given by

$P=x*y$

Putting the value of 'y'

$P=x*(30-x)$

$P=30x-x^2$

This is required expression  

To maximize P

We have to differentiate

$P=30x-x^2$

On differentiation we get

ӘР дх Ә(30х — ?) дх

$\frac{\partial P}{\partial x}=30*1-2*x$

$\frac{\partial P}{\partial x}=30-2x$

When $\frac{\partial P}{\partial x}=0$

$30-2x=0$

$30=2x$

$x=30/2$

$x=15$

From equation (1) we get

$y=30-x$

$y=30-15$

$y=15$

To maximize the product both nuber should be equal

The interval of interest of objective function is

The largest possible product  $=x*y$

$=15*15$

$=225$