Question

What two positive real numbers whose product is 21 have the smallest possible​ sum?

Answer 1

Answer:

$x=\sqrt{21}$

$y=\sqrt{21}$

Step-by-step explanation:

Let x  and y are the two positive real numbers

product of two numbers is 21

$x y=21$

$y=\frac{21}{x}$

The sum of the two numbers is f(x) =x+y

Replace y with 21/x

$f(x) =x+\frac{21}{x}$

we need to find smallest possible sum , so we take derivative using power rule

$f'(x)= 1-\frac{21}{x^2}$

when sum is minimum then the derivative is equal to 0

$0= 1-\frac{21}{x^2}$

$0=\frac{x^2-21}{x^2}$

multiply both sides by x^2

$x^2-21=0$

$x^2=21$

Take square root on both sides

$x=\sqrt{21}$

$y=\frac{21}{x}$

$y=\frac{21}{\sqrt{21}}$

$y=\sqrt{21}$