If all roots of the equation x^3 - 2ax^2 + 3b | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let the roots of the equation be $x_1, x_2, x_3 > 0$.From Vieta's formulas:$x_1 + x_2 + x_3 = 2a$$x_1x_2 + x_2x_3 + x_3x_1 = 3b$$x_1x_2x_3 = 8$Usi

Vedclass · Accepted Answer

$4$

Vedclass · Answer

(A) Let the roots of the equation be $x_1, x_2, x_3 > 0$.From Vieta's formulas:$x_1 + x_2 + x_3 = 2a$$x_1x_2 + x_2x_3 + x_3x_1 = 3b$$x_1x_2x_3 = 8$Using the Arithmetic Mean-Geometric Mean ($AM$-$GM$) inequality for the terms $x_1x_2, x_2x_3, x_3x_1$:$\frac{x_1x_2 + x_2x_3 + x_3x_1}{3} \geq \sqrt[3]{(x_1x_2)(x_2x_3)(x_3x_1)}$$\frac{3b}{3} \geq \sqrt[3]{(x_1x_2x_3)^2}$$b \geq \sqrt[3]{8^2}$$b \geq \sqrt[3]{64}$$b \geq 4$Therefore,the minimum value of $b$ is $4$.

If all roots of the equation $x^3 - 2ax^2 + 3bx - 8 = 0$ are positive,where $a, b \in R$,then the minimum value of $b$ is:

Similar Questions

If $\alpha, \beta$ are the roots of the equation $x^2 - 3x + a = 0$ and $\gamma, \delta$ are the roots of the equation $x^2 - 12x + b = 0$,and $\alpha, \beta, \gamma, \delta$ form an increasing $G.P.$,then $(a, b) = $

The sum to $n$ terms of the series $2^2 + 4^2 + 6^2 + \dots$ is

The sum of all two-digit positive numbers which, when divided by $7$, yield $2$ or $5$ as a remainder is:

The sum to $n$ terms of the series $1 \cdot 3^2 + 2 \cdot 5^2 + 3 \cdot 7^2 + \dots$ is

If the $p^{th}$ term of an $A.P.$ is $q$ and the $q^{th}$ term is $p,$ then its $r^{th}$ term is

Vedclass Test Series

Exam Paper Generator

Online Exam Module