Question

solution of this question

Answer 1

Given  4α/(α² +1) ≥ 1
 
=>   4 / (α + 1/α)  ≥ 1

Let  α + 1/α  = 2 n + 1  as it is an odd integer..   n is some integer.

=>   4 / (2n+1)  ≥  1
=>   2n+1 ≤ 4
=>   n ≤ 3/2

If n is negative the inequality is not valid.  n can be 0 and 1.

So there are two possible values of n. There are two possible values of α + 1/α  which are  1 and 3.

=>  α² - α + 1 = 0         and      α² - 3 α + 1 = 0
=>  α is imaginary .                  α = [3 + √5] /2

So α  has two real values.  If imaginary values can be also taken into account, then α has four values.