2016 AMC 12A Problems/Problem 24

Problem

There is a smallest positive real number $a$ such that there exists a positive real number $b$ such that all the roots of the polynomial $x^3-ax^2+bx-a$ are real. In fact, for this value of $a$ the value of $b$ is unique. What is this value of $b$ ?

$\textbf{(A)}\ 8\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 12$

Solution

$\text{The acceleration must be zero at the }x\text{-intercept: inflection point for minimum }a\text{ value to exist.}$ $x^3-ax^2+bx-a\rightarrow 3x^2-2ax+b\rightarrow 6x-2a\rightarrow x=\frac{a}{3}\text{ for inflection \& intercept}$ $\text{Minimum: }y=\left(x-\frac{a}{3}\right)^3$ $x^3-ax^2+\left(\frac{a^2}{3}\right)x-\frac{a^3}{27}$ $\frac{a^3}{27}=a\rightarrow a^2=27\rightarrow a=3\sqrt{3}$ $b=\frac{a^2}{3}=\frac{27}{3}=\boxed{9}$ $f(x)=x^3-\left(3\sqrt{3}\right)x^2+9x-3\sqrt{3}$ $f(x)=0\rightarrow x=\sqrt{3}\text{ triple root.}$ $\text{Complete the cube (!)}$

2016 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 23	Followed by Problem 25
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2016 AMC 12A Problems/Problem 24

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