Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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Problem C. 1093. (October 2011)

C. 1093. The value of the function f(x) for a real number x is the smallest value out of x2-4x+3, x-1 and -x+7. Determine the number of solutions of the equation f(x)=c, depending on the real parameter c.

(5 pont)

Deadline expired on November 10, 2011.

Sorry, the solution is available only in Hungarian. Google translation

Megoldás. \(\displaystyle x^4x+3=x-1\) akkor, ha \(\displaystyle x=0\) vagy \(\displaystyle x=4\), ugyanakkor \(\displaystyle x^4x+3=-x+7\) akkor, ha \(\displaystyle x=-1\) vagy \(\displaystyle x=4\). Ekkor az \(\displaystyle f(x)\) a következő: ha \(\displaystyle x<1\), akkor \(\displaystyle x-1\); ha \(\displaystyle 1\le x<4\), akkor \(\displaystyle x^2-4x+3\); ha \(\displaystyle 4\le x\), akkor \(\displaystyle -x+7\).

lokális szélsőérték helye értéke
max. 1 0
max. 4 3
min. 2 -1

A megoldások száma:

0 \(\displaystyle c>3\)
1 \(\displaystyle c=3\)
2 \(\displaystyle 0<c<3\) vagy \(\displaystyle c<-1\)
3 \(\displaystyle c=0\) vagy \(\displaystyle c=-1\)
4 \(\displaystyle -1<c<0\)


294 students sent a solution.
5 points:196 students.
4 points:27 students.
3 points:16 students.
2 points:42 students.
1 point:4 students.
0 point:6 students.
Unfair, not evaluated:3 solutions.

Problems in Mathematics of KöMaL, October 2011