Answer:
[tex]x=\sqrt{21}[/tex]
[tex]y=\sqrt{21}[/tex]
Step-by-step explanation:
Let x and y are the two positive real numbers
product of two numbers is 21
[tex]x y=21[/tex]
[tex]y=\frac{21}{x}[/tex]
The sum of the two numbers is f(x) =x+y
Replace y with 21/x
[tex]f(x) =x+\frac{21}{x}[/tex]
we need to find smallest possible sum , so we take derivative using power rule
[tex]f'(x)= 1-\frac{21}{x^2}[/tex]
when sum is minimum then the derivative is equal to 0
[tex]0= 1-\frac{21}{x^2}[/tex]
[tex]0=\frac{x^2-21}{x^2}[/tex]
multiply both sides by x^2
[tex]x^2-21=0[/tex]
[tex]x^2=21[/tex]
Take square root on both sides
[tex]x=\sqrt{21}[/tex]
[tex]y=\frac{21}{x}[/tex]
[tex]y=\frac{21}{\sqrt{21}}[/tex]
[tex]y=\sqrt{21}[/tex]
