If a_1, a_2, ..., a_n are positive real numbe | vedclass

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(A) We use the Arithmetic Mean-Geometric Mean $(AM \geq GM)$ inequality for $n$ positive real numbers.For the set of numbers ${a_1, a_2, ..., a_{n-1},

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$n(2c)^{1/n}$

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(A) We use the Arithmetic Mean-Geometric Mean $(AM \geq GM)$ inequality for $n$ positive real numbers.For the set of numbers ${a_1, a_2, ..., a_{n-1}, 2a_n}$,the $AM \geq GM$ inequality states:$\frac{a_1 + a_2 + ... + a_{n-1} + 2a_n}{n} \geq (a_1 \cdot a_2 \cdot ... \cdot a_{n-1} \cdot 2a_n)^{1/n}$Since the product $a_1 \cdot a_2 \cdot ... \cdot a_n = c$,we substitute this into the inequality:$\frac{a_1 + a_2 + ... + a_{n-1} + 2a_n}{n} \geq (2 \cdot a_1 \cdot a_2 \cdot ... \cdot a_n)^{1/n}$$\frac{a_1 + a_2 + ... + a_{n-1} + 2a_n}{n} \geq (2c)^{1/n}$Multiplying both sides by $n$,we get:$a_1 + a_2 + ... + a_{n-1} + 2a_n \geq n(2c)^{1/n}$Thus,the minimum value is $n(2c)^{1/n}$.

If $a_1, a_2, ..., a_n$ are positive real numbers whose product is a fixed number $c$,then the minimum value of $a_1 + a_2 + ... + a_{n-1} + 2a_n$ is

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