Difference between revisions of "1981 AHSME Problems/Problem 28"
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==Problem 28== | ==Problem 28== | ||
| − | Consider the set of all equations <math> x^3 + a_2x^2 + a_1x + a_0 = 0</math>, where <math> a_2</math>, <math> a_1</math>, <math> a_0</math> are real constants and <math> |a_i| < 2</math> for <math> i = 0,1,2</math>. Let <math> r</math> be the largest positive real number which satisfies at least one of these equations. | + | Consider the set of all equations <math> x^3 + a_2x^2 + a_1x + a_0 = 0</math>, where <math> a_2</math>, <math> a_1</math>, <math> a_0</math> are real constants and <math> |a_i| < 2</math> for <math> i = 0,1,2</math>. Let <math> r</math> be the largest positive real number which satisfies at least one of these equations. Let <math> r</math> be the largest positive real number that satisfies at least one of these equations. Which of the inequalities below does <math> r</math> satisfy? |
<math> \textbf{(A)}\ 1 < r < \dfrac{3}{2}\qquad \textbf{(B)}\ \dfrac{3}{2} < r < 2\qquad \textbf{(C)}\ 2 < r < \dfrac{5}{2}\qquad \textbf{(D)}\ \dfrac{5}{2} < r < 3\qquad \\ \textbf{(E)}\ 3 < r < \dfrac{7}{2}</math> | <math> \textbf{(A)}\ 1 < r < \dfrac{3}{2}\qquad \textbf{(B)}\ \dfrac{3}{2} < r < 2\qquad \textbf{(C)}\ 2 < r < \dfrac{5}{2}\qquad \textbf{(D)}\ \dfrac{5}{2} < r < 3\qquad \\ \textbf{(E)}\ 3 < r < \dfrac{7}{2}</math> | ||
| − | + | ==Solution== | |
| + | |||
| + | Since <math>x^3 = -(a_2x^2 + a_1x + a_0)</math> and <math>x</math> will be as big as possible, we need <math>x^3</math> to be as big as possible, which means <math>a_2x^2 + a_1x + a_0</math> is as small as possible. Since <math>x</math> is positive (according to the options), it makes sense for all of the coefficients to be <math>-2</math>. | ||
| + | |||
| + | Evaluating <math>f(\frac{5}{2})</math> gives a negative number, <math>f(3)</math> 1, and <math>f(\frac{7}{2})</math> a number greater than 1, so the answer is <math>\boxed{D}</math> | ||
| + | |||
| + | ==See also== | ||
| + | |||
| + | {{AHSME box|year=1981|num-b=27|num-a=29}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 14:34, 28 June 2025
Problem 28
Consider the set of all equations 
, where 
, 
, 
are real constants and 
for 
. Let 
be the largest positive real number which satisfies at least one of these equations. Let be the largest positive real number that satisfies at least one of these equations. Which of the inequalities below does satisfy?
Solution
Since 
and 
will be as big as possible, we need 
to be as big as possible, which means 
is as small as possible. Since is positive (according to the options), it makes sense for all of the coefficients to be 
.
Evaluating 
gives a negative number, 
1, and 
a number greater than 1, so the answer is 
See also
| 1981 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 27 |
Followed by Problem 29 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
