What is the minimum value of $p$ and $q$ in the equation $x^{3} - p x^{2} + q x - 8 = 0$ where $p$ and $q$ are positive real numbers and roots of the equation are real?

4, 8

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6,12

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8, 12

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None of these

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Solution

The correct option is B
6,12

$x^{3} - p x^{2} + q x - 8 = 0$

Let the roots be $α, β, γ$

$α + β + γ = p$

$α β γ = 8$

$\frac{α + β + γ}{3} \geq \sqrt[3]{α β γ}$
$p \geq 6$
Minimum value of $p$ is $6.$

This happens when all the roots are equal.
$α = β = γ = 2$
When $p = 6, q = 12$

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