Respuesta :

Let xy = 25, x and y > 0.

[tex]y = \frac{25}{x}[/tex]
[tex]\text{Denote S as: } x + y[/tex]
[tex]S(x) = x + \frac{25}{x}[/tex]

[tex]\text{Minimum: }S'(x) = 0, S''(x) > 0[/tex]
[tex]S'(x) = 1 - \frac{25}{x^{2}}[/tex]
[tex]1 - \frac{25}{x^{2}} = 0[/tex]
[tex]x^{2} = 25[/tex]
[tex]x = 5 (x > 0)[/tex]

[tex]S''(x) = \frac{50}{x^{3}} > 0, (x > 0)[/tex]

Thus, at x = 5, there lies a minimum summation, and this forms a square with side length of 5.

The two positive values whose product is 25 and their sum is a minimum are; +5 and +5.

Let the two numbers be x and y.

We are told that their product is 25

Thus;

xy = 25

y = 25/x

We are told their sum is a minimum.

Thus;

S = x + y

Putting 25/x for y gives;

S = x + (25/x)

Since the sum is minimum, let us find the derivative and equate to zero;

S'(x) = 1 - 25/x²

At S'(x) = 0, we have;

1 - 25/x² = 0

25/x² = 1

x² = 25

Thus;

x = √25

x = 5

Putting 5 for x into y = 25/x;

y = 25/5

y = 5

Thus, the two positive values are 5.

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