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

$$\mathrm{If}\mid z-\frac{4}{z}\mid =2,\mathrm{then}\mathrm{find}\mathrm{the}\mathrm{maximum}$$$$\mathrm{value}\mathrm{of}\mid z\mid .$$

Answered by ajfour last updated on 15/Aug/17

$$\mid (\mid \mathrm{z}\mid -\frac{4}{\mid \mathrm{z}\mid })\mid ⩽2$$$$\mathrm{let}\mid \mathrm{z}\mid =\mathrm{t}$$$$\Rightarrow \mid \mathrm{t}-\frac{4}{\mathrm{t}}\mid ⩽2$$$$\Rightarrow -2⩽\mathrm{t}-\frac{4}{\mathrm{t}}⩽2$$$$\Rightarrow -2\mathrm{t}⩽{\mathrm{t}}^2-4⩽2\mathrm{t}(\mathrm{t}⩾0)$$$$\mathrm{case}\mathrm{I}:$$$${\mathrm{t}}^2+2\mathrm{t}⩾4$$$${(\mathrm{t}+1)}^2⩾5\Rightarrow \mathrm{t}⩾\sqrt{5}-1$$$$\mathrm{case}\mathrm{II}:$$$${\mathrm{t}}^2-2\mathrm{t}⩽4$$$$\Rightarrow {(\mathrm{t}-1)}^2⩽5$$$$\Rightarrow \mathrm{t}⩽\sqrt{5}+1$$$$\mathrm{so}\mid \mathrm{z}{\mid }_{\max }=\sqrt{5}+1.$$

Commented by Tinkutara last updated on 15/Aug/17

$$\mathrm{Thank}\mathrm{you}\mathrm{very}\mathrm{much}\mathrm{Sir}!$$

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