Problem C. 866. (October 2006)

C. 866. Find the value of parameter a such that the distance between two roots of the equation x²-4ax+5a²-6a=0 is the greatest.

(5 pont)

Deadline expired on November 15, 2006.

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

Megoldás. Az egyenlet két gyökét x₁-gyel és x₂-vel jelölve keressük |x₁-x₂| maximumát.

$x_{1,2}=\frac{4a\pm\sqrt{24a-4a^2}}{2}.$

Innen

$|x_1-x_2|=\sqrt{24a-4a^2}.$

A négyzetgyökfüggvény szigorú monotonitása miatt a jobb oldali kifejezés akkor maximális, amikor a gyökjel alatti kifejezés is az, vagyis ha 24a-4a²=4a(6-a) maximális, tehát $a=\frac{0+6}{2}=3$ esetén.

A két gyök a=3 esetén lesz legmesszebbre egymástól.

Statistics:

471 students sent a solution.

5 points: 174 students.

4 points: 128 students.

3 points: 76 students.

2 points: 27 students.

1 point: 33 students.

0 point: 28 students.

Unfair, not evaluated: 5 solutionss.

Problems in Mathematics of KöMaL, October 2006

471 students sent a solution.
5 points:	174 students.
4 points:	128 students.
3 points:	76 students.
2 points:	27 students.
1 point:	33 students.
0 point:	28 students.
Unfair, not evaluated:	5 solutionss.

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