1981 AMC 12 Problem 28

Question

Consider the set of all equations

x^3+a_2x^2+a_1x+a_0=0,

where

a_2,

a_1,

a_0

are real constants and

|a_i|\le2

for

i=0,

1,

2.

Let

r

be the largest positive real number which satisfies at least one of these equations. Then

1\le r\lt\frac32

\frac32\le r\lt2

2\le r\lt\frac52

\frac52\le r\lt3

3\le r\lt\frac72

For positive

x,

the coefficients that push a root farthest right take their lower bounds

Bracket the largest root of

x^3-2x^2-2x-2

at two consecutive choice endpoints

Write an allowed polynomial as

g.

For

x\ge0,

\begin{aligned} g(x)&=x^3+a_2x^2\\ &\quad+a_1x+a_0\\ &\ge f(x), \end{aligned}

where

f(x)=x^3-2x^2-2x-2.

Thus no positive root can exceed the largest root

\rho

of

f.

Taking

a_2=a_1=a_0=-2

gives

g=f,

so

\rho

itself is attained. Since

\begin{gathered} f\left(\frac52\right)=-\frac{31}{8}\lt0,\\ f(3)=1\gt0, \end{gathered}

we have

\frac{5}{2}\lt\rho\lt3.

Therefore, the correct answer is D.

Problem 28 in other years