Question

Equation $x^4+a{x}^3+b{x}^2+cx+1=0$ has real roots ($a,b,c$ are non-negative). Maximum value of $a$ is

• A. $10$
• B. $9$
• C. $5$
• D. $-4$

Answer 1

The correct option is D $-4$

let ${\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4$ be the roots of the given equation $x^4+a{x}^3+b{x}^2+cx+1=0$

Sum of the roots of the given polynomial is ${\alpha }_1+{\alpha }_2+{\alpha }_3+{\alpha }_4=\frac{-a}{1}=-a$

product of the roots of the given polynomial is ${\alpha }_1\times {\alpha }_2\times {\alpha }_3\times {\alpha }_4=\frac{(-1{)}^4\times 1}{1}=1$

We know that, $A.M.\ge G.M.$

therefore,

$\frac{{\alpha }_1+{\alpha }_2+{\alpha }_3+{\alpha }_4}{4}\ge \sqrt[4]{4\alpha 1\times \alpha 2\times \alpha 3\times \alpha 4}$

$⟹\frac{-a}{4}\ge \sqrt[4]{41}$

$⟹-a\ge 4$

$⟹a\le -4$

from this $a$ value is always negative with a maximum of $-4$