Math help me ?

300 = x * y
s = x + 3y

We want to minimize s

300/x = y

s = x + 3 * (300/x)
s = x + 900 * x^(-1)

Derive s with respect to x

ds/dx = 1 + 900 * (-1) * x^(-2)

Let ds/dx = 0.  The tells us when the rate of change for s with respect to the rate of change for x is at a critical point

0 = 1 - 900 / x^2

Solve for x

900/x^2 = 1
900 = x^2
x = -30 , 30
y = 300/x
y = -10 , 10

x and y are positive

x = 30 , y = 10

30 * 10 = 300

30 + 3 * 10 = 30 + 30 = 60

The minimum sum of x + 3y is 60

Let x be the first number.
Then the second number = 300/x

The sum of the first number plus three times the second number
= x + 3(300/x)
= x + (900/x)
= (x² + 900)/x
= (x² + 30²)/x
= [(x² - 60x + 30²) + 60x]/x
= [(x - 30)² + 60x]/x
= [(x - 30)²/x] + 60 ≥ 60
because [(x - 30)²/x] ≥ 0 for any positive number x.

The smallest sum of the first number plus three times the second number = 60

Do they have to be integers, or can they be fractions?

Actually, scratch that.  It doesn't matter.

For such a question, the minimal sum of two factors is achieved when the factors are the same.  In this case, when one of the factors is multiplied by some number "n", the minimal sum is when the first factor is "n" times the second.

Like this:
XY=300
X+3Y=Z

Z is minimal when X=3Y

Substituting, we get:
3Y*Y = 300
3Y^2 = 300
Y^2 = 100
Y = 10

X+3Y = 30 + 30 = 60

X = 30, Y = 10

Let the two numbers be a and b respectively. If a + 3b is to be minimized, likely a > b. But let's see about that; I'm going to first list the factors of 300; I'll do these in pairs.

1, 300
2, 150
3, 100
4, 75
5, 60
6, 50
10, 30
12, 25
15, 20

The first 3 can immediately be ignored as the best you can do is 100 + 3(3) = 109 as the lowest sum.

The next 3 you get as far down as 68 (50 + 3(6) = 68).

10 and 30...10 + 3(30) = 100, no. Or: 30 + 3(10) = 60. Lowest so far.

12 and 25...12 + 3(25) = 87, no. Or: 25 + 3(12) = 61. No.

15 and 20...15 + 3(20) = 75, no. Or: 20 + 3(15) = 65. No.

Lowest sum you can get is 60 if the first number is 30 and the second 10.

xy = 300, so y = 300/x
z = x + 3y = x + 3(300/x) = x + 900x^(-1)
dz/dx = 1 - 900x^(-2)
1 - 900x^(-2) = 0
900x^(-2) = 1
900x^2 = 1/1 = 1
x^2 = 1/900
x = +/- sqrt(1/900) = +/- 1/30
If x = 1/30 then y = 300/(1/30) = 9000, so z = (1/30) + 3*9000 = 27000+(1/30)
If x = -1/30 then y = 300/(-1/30) = -9000, so z = (-1/30) + 3(-9000) = -27000-(1/30)
Answer: -1/30, -9000

Let the numbers be 'm' & 'n'
Hence
mn = 300
m + 3n = least
Hence
Substitute
m (- m )=300
m^2 - 300
m = sqrt(300))
m = 17.32....
n = 17.32,,,