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 .