If-z-4-z-2-then-find-the-maximum-value-of-z-

Question Number 19690 by Tinkutara last updated on 14/Aug/17

If ∣ z - \frac{4}{z} ∣ = 2, then find the maximum

value of ∣ z ∣ .

Answered by ajfour last updated on 15/Aug/17

∣ (∣ z ∣ - \frac{4}{∣ z ∣}) ∣⩽ 2

let ∣ z ∣= t

\Rightarrow ∣ t - \frac{4}{t} ∣⩽ 2

\Rightarrow - 2 ⩽ t - \frac{4}{t} ⩽ 2

\Rightarrow - 2 t ⩽ t^{2} - 4 ⩽ 2 t (t ⩾ 0)

case I :

t^{2} + 2 t ⩾ 4

{(t + 1)}^{2} ⩾ 5 \Rightarrow t ⩾ \sqrt{5} - 1

case II :

t^{2} - 2 t ⩽ 4

\Rightarrow {(t - 1)}^{2} ⩽ 5

\Rightarrow t ⩽ \sqrt{5} + 1

so ∣ z ∣_{\max} = \sqrt{5} + 1 .

Commented by Tinkutara last updated on 15/Aug/17

Thank you very much Sir!