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Difference between revisions of "1995 AHSME Problems/Problem 12"

(New page: ==Problem== Let <math>f</math> be a linear function with the properties that <math>f(1) \leq f(2), f(3) \geq f(4),</math> and <math>f(5) = 5</math>. Which of the following is true? <math...)
 
(→‎Solution)
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==Solution==
 
==Solution==
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A linear function has the property that <math>f(a)\leq f(b)</math> either for all a<b, or for all b<a. Since <math>f(3)\geq f(4)</math>, <math>f(1)\geq f(2)</math>. Since f(1)\leq f(2)<math>, </math>f(1)=f(2)<math>. And if </math>f(a)=f(b)<math> for a≠b, then f(x) is a constant function. Since </math>f(5)=5<math>, </math>f(0)=5\Rightarrow \mathrm{(D)}$
+
A linear function has the property that <math>f(a)\leq f(b)</math> either for all a<b, or for all b<a. Since <math>f(3)\geq f(4)</math>, <math>f(1)\geq f(2)</math>. Since <math>f(1)\leq f(2)</math>, <math>f(1)=f(2)</math>. And if <math>f(a)=f(b)</math> for a≠b, then f(x) is a constant function. Since <math>f(5)=5</math>, <math>f(0)=5\Rightarrow \mathrm{(D)}</math>
  
 
==See also==
 
==See also==

Revision as of 10:00, 9 January 2008

Problem

Let $f$ be a linear function with the properties that $f(1) \leq f(2), f(3) \geq f(4),$ and $f(5) = 5$. Which of the following is true?


$\mathrm{(A) \ f(0) < 0 } \qquad \mathrm{(B) \ f(0) = 0 } \qquad \mathrm{(C) \ f(1) < f(0) < f( - 1) } \qquad \mathrm{(D) \ f(0) = 5 } \qquad \mathrm{(E) \ f(0) > 5 }$

Solution

A linear function has the property that $f(a)\leq f(b)$ either for all a<b, or for all b<a. Since $f(3)\geq f(4)$, $f(1)\geq f(2)$. Since $f(1)\leq f(2)$, $f(1)=f(2)$. And if $f(a)=f(b)$ for a≠b, then f(x) is a constant function. Since $f(5)=5$, $f(0)=5\Rightarrow \mathrm{(D)}$

See also

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