Maximize it!

Algebra Level 4

$\large a^2b^3c^4$

Positive real numbers $a$ , $b$ and $c$ are such that $a+b+c=18$ . If the maximum value of expression above can be written as $p_1^\alpha \cdot p_2^\beta$ , where $p_1$ and $p_2$ are prime numbers and $\alpha$ , $\beta$ are positive integers, find $p_1+p_2+\alpha+\beta$ .

The answer is 27.

1 solution

Sabhrant Sachan
Oct 1, 2017

Relevant wiki: Arithmetic Mean - Geometric Mean

By A.M-G.M

$\dfrac{ \left( \frac{a}{2}+\frac{a}{2} \right) + \left( \frac{b}{3}+\frac{b}{3}+\frac{b}{3} \right) +\left( \frac{c}{4}+\frac{c}{4}+\frac{c}{4}+\frac{c}{4} \right) }{9} \ge \left( \dfrac{a^2 b^3 c^4}{2^2 3^3 4^4 }\right)^{\frac{1}{9}} \\ 2^9 \ge \left( \dfrac{a^2 b^3 c^4}{2^2 3^3 4^4 }\right) \\ a^2 b^3 c^4 \le 3^3 2^{19 }$

and for this maximum to exist $\dfrac{a}{2} = \dfrac{b}{3} = \dfrac{c}{4}$

$a=4$ , $b=6$ , $c=8$

Its simply weighted AM-GM

Md Zuhair - 3 years, 8 months ago

Maximize it!

The answer is 27.

1 solution

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