Find a positive number such that the sum of r times its reciprocal and t times the number is as small as possible.

Question

ilovecake · Answer

1/x

anonymous · Answer

im using online homework and i put that in and it didnt work says wrong still :/

ilovecake · Answer

:(

ranga · Answer

Let the number be x and let the sum be y.
Sum y = "r times its reciprocal and t times the number"
y = r/x + tx
Sum should be as small as possible.  
To minimize y, find dy/dx, equate to zero and solve for x. (r and t are constants).

anonymous · Answer

still cant get it lol to many variables

ranga · Answer

You can treat r and t as constants.
y = r/x + tx = rx^-1 + tx
Find the derivative
dy/dx = -rx^-2 + t
Equate dy/dx to zero and solve for x.
-r/x^2 + t = 0
t = r/x^2
x^2 = r/t
x = sqrt(r/t)

anonymous · Answer

thanks man :)

ranga · Answer

$$\Large x = \sqrt{\frac{ r }{ t }}$$

ranga · Answer

You are welcome.