One number is 6 less than 4 times a second number. Find a pair of such numbers so that their product is as small as possible.

Question

One number is 6 less than 4 times a second number.Find a pair of such numbers so that their product is as small as possible.

these two numbers are:

the smallest possible product is:

Answer 1

x=4y-6

product = xy = (4y-6)y = 4y^2 -6y

take the derivative and set = 0

8y - 6 = 0

8y = 6

y = 6/8 = 3/4

x = 4(3/4) -6 = 3-6 = -3

x=-3, y =3/4, xy = -9/4 is a minimum product of xy

y=1 means x = 4(1)-6 = -2 and xy =-2 which is > -9/4

y=1/2 means x =4(1/2)-6 =-4 and xy=-2> -9/4

xy gets larger if y gets larger or smaller than 3/4

the 2nd derivative=8 >0 which means the solution is a local minimum. It's also a global minimum