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

$$\mathrm{The}\mathrm{natural}\mathrm{number}\mathrm{n}\mathrm{for}\mathrm{which}$$$$\mathrm{the}\mathrm{expression}\mathrm{y}={5\log }^2{}_3\left(\mathrm{n}\right)-$$$$\log {}_3\left({\mathrm{n}}^{12}\right)+9,\mathrm{has}\mathrm{the}\mathrm{minimum}$$$$\mathrm{value}\mathrm{is}\mathrm{\_}\mathrm{\_}\mathrm{\_}$$

Commented by bobhans last updated on 08/Jun/20

$$\mathrm{let}\log {}_3\left(\mathrm{n}\right)=\mathrm{x}\Rightarrow \mathrm{y}={5\mathrm{x}}^2-12\mathrm{x}+9$$$${\mathrm{y}}_{\min }\mathrm{at}\mathrm{x}=-\frac{\mathrm{b}}{2\mathrm{a}}=\frac{6}{5}.\mathrm{here}\log {}_3\left(\mathrm{n}\right)=\frac{6}{5}$$$$\Rightarrow \mathrm{n}=3^{\frac{6}{5}}\equiv 3.70\mathrm{which}\mathrm{is}\mathrm{not}\mathrm{natural}$$$$\mathrm{number}.\mathrm{Hence}\mathrm{minimum}\mathrm{occurs}\mathrm{at}\mathrm{the}$$$$\mathrm{clossest}\mathrm{integer}.\mathrm{now}4>3^{\frac{6}{5}}$$$$\Rightarrow 1024>729$$

Commented by bemath last updated on 08/Jun/20

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

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