Question

a. The sum of a number and it’s reciprocal is 25/12 What is the number?

b. Prove that the sum of a positive number and it’s reciprocal is always greater than or equal to 2.

Answer 1

b. Let x be positive

\(x+\frac{1}{x}=a;\)

\(\frac{x^2+1}{x}=a\)

ie. x² - ax + 1 = 0

\(x=\frac{a\pm \sqrt{a^2-4}}{2}\)

discriminant must be greater than or equal to zero

a² - 4 > 0 ; a2 > 4

∴ a > 2