The-natural-number-n-for-which-the-expression-y-5log-2-3-n-log-3-n-12-9-has-the-minimum-value-is-

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

The natural number n for which

the expression y = {5 \log}^{2}_{3} (n) -

\log_{3} (n^{12}) + 9, has the minimum

value is___

Commented by bobhans last updated on 08/Jun/20

let \log_{3} (n) = x \Rightarrow y = {5 x}^{2} - 12 x + 9

y_{\min} at x = - \frac{b}{2 a} = \frac{6}{5} . here \log_{3} (n) = \frac{6}{5}

\Rightarrow n = 3^{\frac{6}{5}} \equiv 3.70 which is not natural

number . Hence minimum occurs at the

clossest integer . now 4 > 3^{\frac{6}{5}}

\Rightarrow 1024 > 729

Commented by bemath last updated on 08/Jun/20

thank you

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