optimization problem?

Question

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

Answer 1

So you are minimizing the function

f(x) = x + (1/x)
"This is the sum of a number x and its reciprocal".

Derivative of f(x) = 1 + (-1/x^2)

Set it equal to zero and you get 0 = 1 + (-1/x^2)

Solve for x and you find that x = 1

This is a calculus problem right.. if not.. ill solve it without using calculus.

Just look at the function
f(x) = x + (1/x)

This is a parabola, if you look at the parabola, its vertex is at x = 1. Therefore the minimum value is at x = 1.

Answer 2

1

1 + 1/1 = 2
2 + 1/2 = 2.5
3 + 1/3 = 3.33

Answer 3

Let x be the number.
y = x + 1/x
dy/dx = 1 - 1/x^2 = 0
1/x^2 = 1
x^2 = 1
x = ± 1
Discarding the negative solution,
x = 1
Guess-and-check might be quicker for this one. . . .