csat-suneung 2005 Q24

csat-suneung · South-Korea · csat__math-science 4 marks Stationary points and optimisation Count or characterize roots using extremum values
The cubic equation in $x$, $\frac { 1 } { 3 } x ^ { 3 } - x = k$, has three distinct real roots $\alpha$, $\beta$, $\gamma$. For a real number $k$, let $m$ be the minimum value of $| \alpha | + | \beta | + | \gamma |$. Find the value of $m ^ { 2 }$. [4 points] 
The cubic equation in $x$, $\frac { 1 } { 3 } x ^ { 3 } - x = k$, has three distinct real roots $\alpha$, $\beta$, $\gamma$. For a real number $k$, let $m$ be the minimum value of $| \alpha | + | \beta | + | \gamma |$. Find the value of $m ^ { 2 }$. [4 points]
Paper Questions
Q1 Q2 Q3 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Q14 Q15 Q16 Q17 Q18 Q19 Q20 Q21 Q22 Q23 Q24 Q25 Q26 (Discrete Mathematics) Q26 (Probability and Statistics) Q27 (Discrete Mathematics) Q27 (Probability and Statistics) Q28 (Discrete Mathematics) Q28 (Probability and Statistics) Q29 (Discrete Mathematics) Q29 (Probability and Statistics) Q30 (Calculus and Differentiation) Q30 (Discrete Mathematics) Q30 (Probability and Statistics)