Question

find two positive numbers whose product is 25 and whose sum is a minimum. (enter your answers as a comma-separated list.)

Answer 1

The required positive number whose product is 25 and whose sum is minimum is 5, 5

Sum of Two numbers

Sum of two numbers means adding two same or different number together

Let two positive numbers xy = 25

Divide both side by x

y  = $\frac{25}{x}$  ----------Equation (1)

Sum of two positive numbers s = x + y  -------Equation (2)

Substitute equation (1) into equation (2)

S = x + $\frac{25}{x}$

differentiate with respect to x

$\frac{ds}{dx} = \frac{d}{dx} (x + \frac{25}{x} ) = 1 - \frac{25}{x^{2} }$

$\frac{ds}{dx} = 1-\frac{25}{x^{2} }$  ---------- Equation (3)

$\frac{ds}{dx} = 0$

$1$$- \frac{25}{x^{2} }$ = 0  ⇒ $x^{2}$ = 25

x = 5

Differentiate equation (3) again

$\frac{d^{2}s }{dx^{2} } = 0 + \frac{50}{x^{2} }$

hence, at point x = 5,   $\frac{d^{2} s}{dx^{2} } = \frac{50}{5^{2} }$  $= \frac{50}{25} = 2$

Value of s minimum at point x = 5.

substitute the value of x into equation (1)

$y = \frac{25}{x} = \frac{25}{5} = 5$

Therefore the required positive numbers are 5, 5.

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