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Question Number 97501 by bemath last updated on 08/Jun/20

The natural number n for which the expression y = 5log^2 _3 (n) − log _3 (n^(12) )+9 , has the minimum value is ___

$$\mathrm{The}\:\mathrm{natural}\:\mathrm{number}\:\mathrm{n}\:\mathrm{for}\:\mathrm{which}\: \\ $$$$\mathrm{the}\:\mathrm{expression}\:\mathrm{y}\:=\:\mathrm{5log}^{\mathrm{2}} \:_{\mathrm{3}} \left(\mathrm{n}\right)\:− \\ $$$$\mathrm{log}\:_{\mathrm{3}} \left(\mathrm{n}^{\mathrm{12}} \right)+\mathrm{9}\:,\:\mathrm{has}\:\mathrm{the}\:\mathrm{minimum} \\ $$$$\mathrm{value}\:\mathrm{is}\:\_\_\_ \\ $$

Commented by bobhans last updated on 08/Jun/20

let log _3 (n)=x ⇒y=5x^2 −12x+9 y_(min) at x = −(b/(2a)) = (6/5). here log _3 (n)=(6/5) ⇒n=3^(6/5) ≡ 3.70 which is not natural number. Hence minimum occurs at the clossest integer . now 4 > 3^(6/5) ⇒1024 > 729

$$\mathrm{let}\:\mathrm{log}\:_{\mathrm{3}} \left(\mathrm{n}\right)=\mathrm{x}\:\Rightarrow\mathrm{y}=\mathrm{5x}^{\mathrm{2}} −\mathrm{12x}+\mathrm{9} \\ $$$$\mathrm{y}_{\mathrm{min}} \:\mathrm{at}\:\mathrm{x}\:=\:−\frac{\mathrm{b}}{\mathrm{2a}}\:=\:\frac{\mathrm{6}}{\mathrm{5}}.\:\mathrm{here}\:\mathrm{log}\:_{\mathrm{3}} \left(\mathrm{n}\right)=\frac{\mathrm{6}}{\mathrm{5}} \\ $$$$\Rightarrow\mathrm{n}=\mathrm{3}^{\frac{\mathrm{6}}{\mathrm{5}}} \:\equiv\:\mathrm{3}.\mathrm{70}\:\mathrm{which}\:\mathrm{is}\:\mathrm{not}\:\mathrm{natural} \\ $$$$\mathrm{number}.\:\mathrm{Hence}\:\mathrm{minimum}\:\mathrm{occurs}\:\mathrm{at}\:\mathrm{the} \\ $$$$\mathrm{clossest}\:\mathrm{integer}\:.\:\mathrm{now}\:\mathrm{4}\:>\:\mathrm{3}^{\frac{\mathrm{6}}{\mathrm{5}}} \\ $$$$\Rightarrow\mathrm{1024}\:>\:\mathrm{729}\: \\ $$

Commented by bemath last updated on 08/Jun/20

thank you

$$\mathrm{thank}\:\mathrm{you} \\ $$

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