Let p- q- r- such that 27pqr-p-q-r3 and 3p-4q-5r-12- Then the value of p-q-r is

Since p, q, r∈ℝ^+ Applying A.M. ≥ G.M. ⇒p+q+r3≥(pqr)^1/3 ⇒ (p+q+r)^3≥27pqr But given, (p+q+r)^3≤27pqr which holds good if (p+q+r)^3=27pqr i.e., A.M. = G.M. ⇒ p= ...

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Byju's Answer

Standard XIII

Mathematics

AM,GM,HM Inequality

Let p, q, r∈ℝ...

Question

Let $p, q, r \in R^{+}$ such that $27 p q r \geq (p + q + r)^{3}$ and $3 p + 4 q + 5 r = 12.$ Then the value of $p + q + r$ is

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Solution

The correct option is A $3$
Since $p, q, r \in R^{+}$
Applying A.M. $\geq$ G.M.
$\Rightarrow \frac{p + q + r}{3} \geq (p q r)^{1 / 3}$
$\Rightarrow (p + q + r)^{3} \geq 27 p q r$
But given, $(p + q + r)^{3} \leq 27 p q r$
which holds good if $(p + q + r)^{3} = 27 p q r$
i.e., A.M. $=$ G.M.
$\Rightarrow p = q = r$

Since $3 p + 4 q + 5 r = 12$
$\Rightarrow 12 p = 12 \Rightarrow p = 1 \Rightarrow q = r = 1$
$∴ p + q + r = 3$

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Let p, q, r∈R+ such that 27pqr≥(p+q+r)3 and 3p+4q+5r=12. Then the value of p+q+r is

The correct option is A 3Since p, q, r∈R+ Applying A.M. ≥ G.M. ⇒p+q+r3≥(pqr)1/3 ⇒(p+q+r)3≥27pqr But given, (p+q+r)3≤27pqr which holds good if (p+q+r)3=27pqr i.e., A.M. = G.M. ⇒p=q=r Since 3p+4q+5r=12 ⇒12p=12⇒p=1⇒q=r=1 ∴ p+q+r=3

Let $p, q, r \in R^{+}$ such that $27 p q r \geq (p + q + r)^{3}$ and $3 p + 4 q + 5 r = 12.$ Then the value of $p + q + r$ is