let-a-1-a-2-a-n-be-positive-real-numbers-such-that-a-1-a-2-a-n-1-then-find-maximum-value-of-a-1-a-1-a-2-a-2-a-n-a-n-

Question Number 145073 by gsk2684 last updated on 02/Jul/21

let a_{1}, a_{2}, \dots, a_{n} be positive

real numbers such that

a_{1} + a_{2} + \dots + a_{n} = 1 then find

maximum value of

a_{1}^{a_{1}} . a_{2}^{a_{2}} \dots . a_{n}^{a_{n}} ?

Commented by shuvam last updated on 02/Jul/21

a r e y o u t e a c h e r ?

Commented by gsk2684 last updated on 02/Jul/21

yes

Answered by MJS_new last updated on 02/Jul/21

\lim_{r \to 0} r^{r} = 1

\Rightarrow let a_{1} \to 1 \land a_{2} = a_{3} = \dots = a_{n} \to 0 \Rightarrow

\frac{1}{n} ⩽ a_{1}^{a_{1}} a_{2}^{a_{2}} \dots a_{n}^{a_{n}} < 1

Commented by gsk2684 last updated on 02/Jul/21

thank you

hou could you write minimum as \frac{1}{n}

kindly explain sir

Commented by MJS_new last updated on 02/Jul/21

minimum at a_{1} = a_{2} = \dots = a_{n} = \frac{1}{n}

\Rightarrow

a_{k}^{a_{k}} = {(\frac{1}{n})}^{\frac{1}{n}} = \frac{1}{\sqrt[n]{n}}

\underset{k = 1}{\prod^{n}} \frac{1}{\sqrt[n]{n}} = {(\frac{1}{\sqrt[n]{n}})}^{n} = \frac{1}{n}

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