Let the numbers are x and y
And it is given that the difference is 180
So we have
[tex] x-y=180 [/tex]
[tex] x = 180+y [/tex]
And let the product is p. So we have
[tex] p = xy [/tex]
Substituting the value of x , we will get
[tex] p = y(180+y) = 180y +y^2 [/tex]
Differentiating p,
[tex] p'= 180+2y [/tex]
Again differentiating ,
[tex] p'' = 2>0 [/tex]
SO we will get minimum.
And to find the minimum, we have to set p' equal to 0 and solve for y.
[tex] 180+2y=0 =>y=-90 [/tex]
Back substituting the value of y, we will get
[tex] x =180+(-90) [/tex]
[tex] x =90 [/tex]
So the numbers are 90 and -90 .
