Question

GIVEN
A > 0
B > 0
f[x] = A x + B/x .
Find the minima for all positive x's

Answer 1

I got this far... and got stuck... Im not sure what to do next, or if I'm on the wrong track altogether.
Find the derivative of f[x]
f[x] = A x + B/x A and B are constants.
= A + B x^-1
f'[x] = A -B x^-2

Find the zeros of the derivative
f'[x] = 0
0 = A -B/x^2

B/ x^2 = A
B = A x^2
B = A x x
B/A = x x
Sqrt[B/A] = Sqrt[x x]
Sqrt[B]/Sqrt[A] = x

Assign that x as the minima,

Answer 2

looks good

several suggestions:

1. for small x, f(x) = B/x, for large x f(x) = Ax. so there should be some sort of transition in the shape of the line. ie a stationary point. you could sketch it to visualise...
2. look at f''(x) and how it behaves across the range of the function.

IOW you have to be sure the stationary point is a max/min and not a saddle / asymtotpe. f'' and a sketch will help.

Answer 3

Like Irish said you dont know if its a minima or a maxima yet

Answer 4

It happens to be in this case, but in future, you need to do take the 2nd derivative, or check the behavior around your critical point

Answer 5

your work looks good

Answer 6

Does anyone else have trouble with the backspace key in the equation editor causing a page refresh? .. sorry in advance if I dont use it for now.

Im trying to feed the the Sqrt[B]/Sqrt[A] into the 2nd derivative to get a confirm on the minima, but having some issues with the algebra..

Do you guys have any tricks for Simplifying
( B)/ (Sqrt[B]/Sqrt[A])

I think it should end up as A*B
But I'm not remembering the rules for making that happen.