Roots of Cubic Equation

Algebra Level 5

For a real number $k$ , the cubic equation $\frac{1}{3}x^3-x=k$ has three distinct roots $\alpha, \beta$ and $\gamma$ . If $m=\min_{k\in \mathbb{R}}(|\alpha|+|\beta|+|\gamma|),$ what is $m^2?$

The answer is 12.

2 solutions

Akif Khan
May 3, 2014

$Applying\quad AM,GM\quad inequality\quad to\quad above\quad expression\\ \Rightarrow \frac { \quad \left| \alpha \right| +\left| \beta \right| +\left| \gamma \right| }{ 3 } \ge \sqrt [ 3 ]{ \left| \alpha \beta \gamma \right| } \\ \Rightarrow \left| \alpha \right| +\left| \beta \right| +\left| \gamma \right| \ge \quad 3\sqrt [ 3 ]{ \left| 3k \right| } \\ Now,\quad above\quad expression\quad will\quad be\quad minimum\quad when\quad k=0,\quad \\ Putting\quad k=0,\quad in\quad cubic\quad equation\quad we\quad get\quad roots\quad \alpha =0,\beta =\sqrt { 3 } ,\gamma =-\sqrt { 3 } \\ Therefore,\quad { m }^{ 2 }=12$

All that you have proved is that the RHS of the inequality attains it's minimum at $k = 0$ . You still have to prove that the LHS also attains it's minimum at $k = 0$ .

Yeshwanth Cherapanamjeri - 7 years ago

bahut badiya bete ;-)

pulkit gupta - 7 years, 1 month ago

Ankit Gupta
Apr 17, 2014

solve when k=0

You have to prove that the minimum of the sum will attain when $k=0$ .

Jit Ganguly - 7 years, 1 month ago

Roots of Cubic Equation

The answer is 12.

2 solutions

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