Optimization problem

Question

Find two positive numbers that satisfy the given requirements.

The second number is the reciprocal of the first number and the sum is a minimum.

first number=?

second number=?

any guidance will be much appreciated!

thank you in advance

Answer 1

f(x) = x + 1/x

take the derivative and set equal to zero

f'(x) = 1 -1/x^2 = 0

multiply by x^2

x^2-1 - 0

x^2 =1

x=sqr1 = 1 (ignore the negative square root)

1/x = 1/1 = 1

the numbers are 1 and 1

they sum to 2

try any other numbers

2 + 1/2 = 2.5

3 +1/3 = 3 1/3

4 + 1/4 = 4.25

3/2 +2/3 = 13/6 = 2 1/6

1.01 + 1/1.01 = 2.000099...

you'll always get more than 2 for the sum

1 and 1 give the smallest sum

Answer 2

F(x)=x+1/x

Then minimizing,

0=F'(x)=1-1/x^2

Mmultiplies by x^2...

X^2-1=0

X=1 or x=-1 <---disqualified since they are positive

So x=1 and the min sum is 2