First slide

Means - A.M/G.M/H.M

Question

If $a + 2 b + 3 c = 12, (a, b, c \in R^{+})$ ,then ${ab}^{2} c^{3}$ is

Moderate

A

$\geq 2^{3}$
B

$\geq 2^{6}$
C

$\leq 2^{6}$
D

None of these

Solution

Given that, $a + 2 b + 3 c = 12$
and a, b, c are positive real numbers
Now, $AM \geq GM$
$\begin{array}{ll} \Rightarrow \frac{a + b + b + c + c + c}{6} \geq \sqrt[6]{{ab}^{2} c^{3}} \\ \Rightarrow \frac{a + 2 b + 3 c}{6} \geq \sqrt[6]{{ab}^{2} c^{3}} \Rightarrow {ab}^{2} c^{3} \leq 2^{6} \end{array}$

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