Question

Find two numbers whose difference is 180 and whose product is a minimum

Answer 1

Let the numbers are x and y

And it is given that the difference is 180

So we have

`x-y=180`

`x = 180+y`

And let the product is p. So we have

`p = xy`

Substituting the value of x , we will get

`p = y(180+y) = 180y +y^2`

Differentiating p,

`p'= 180+2y`

Again differentiating ,

`p'' = 2>0`

SO we will get minimum.

And to find the minimum, we have to set p' equal to 0 and solve for y.

`180+2y=0 =>y=-90`

Back substituting the value of y, we will get

`x =180+(-90)`

`x =90`

So the numbers are 90 and -90 .