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

### Statistics:

 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