Prove that for all positive real numbers x, the sum of x and its reciprocal is greater than or equal to 2

Question

Mathematics

Arnav Stephenson

7 May, 11:40

Prove that for all positive real numbers x, the sum of x and its reciprocal is greater than or equal to 2

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Dalia · Answer 1

Let f (x) = x + (1/x) for any positive real x

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

f'' (x) = 1 / (x^3)

Set f' (x) = 0, we have x=1

f'' (1) = 1 >0

Hence, the minimum value of f (x) is f (1) = 2