Answer :
Minimum number is 33+4*6= 57
Give us two positive numbers, x and y.
196 is the result: x*y = 19
The minimum is the product of the first plus four times the second: x + 4y.
The first equation results in y = 196/x. Put that into the second equation as a replacement:
x + 4y = x + 4(196/x) = x + 784/x
Taking the first derivative now, we can solve for x by setting it to zero.
d(x + 784/x)/dx = 1 - 784/[tex]x^{2}[/tex]
0 = 1 - 784/[tex]x^{2}[/tex]
0 = [tex]x^{2}[/tex]- 784 [[tex]x^{2}[/tex] multiplied on both sides]
784 = [tex]x^{2}[/tex]
[tex]\sqrt{784}[/tex] = [tex]\sqrt{x^{2} }[/tex]
Y = 196/x = 7 since we want a positive integer, hence x = 28
The factors of 196 are 196*1, 99*2, 66*3, 6*33,22*19, and 11*18 as a check.
196 + 4*1 = 200
99 + 4*2 = 107
66 + 4*3 = 78
33+4*6= 57
22 + 4*19 = 98
11 + 4*18=82
In fact, 33+4*6= 57 is minimum.
To learn more about factors:
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