Question

If l, m, n are 3 positive roots of the equation (p, q $ϵ$ R), then the minimum value of l + 2m + 3n is

• A.
• B.
• C. 18
• D. 6

Answer 1

The correct option is C

18

Consider l, 2m, 3n and use AM $\ge$ GM

$\frac{l+2m+3n}{3}$ $\ge$ ${\left(l.2m.3n\right)}^{\frac{1}{3}}$

l + 2m + 3n $\ge$ 3 ${\left(6lmn\right)}^{\frac{1}{3}}$ -----------(1)

Given l, m, n are roots of $x^3$ - p$x^2$ + qx - 36 = 0

Product of roots = lmn = 36

(1) $\Rightarrow$ l + 2m + 3n $\ge$ 3${(6\times 36)}^{\frac{1}{3}}$

$\ge 3.6$

l + 2m + 3n $\ge$ 18

so, minimum value of l + 2m + 3n is 18