Problem C. 873. (November 2006)

C. 873. For what real values of x will the value of the expression $\sqrt{2\sin x}-\sin x$ be a maximum?

(5 pont)

Deadline expired on December 15, 2006.

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

Megoldás. $f(x)=\sqrt{2\sin x}-\sin x$ . Legyen $a=\sqrt{2\sin x}$ , $g(x)=a-\frac{1}{2}a^2=a(1-\frac{1}{2}a)$ . Ez egy másodfokú függvény, maximumát a=1-ben veszi fel. Ekkor $a=\sqrt{2\sin x}=1$ , vagyis sin x=1/2. Tehát f(x) értéke $x=\frac{\pi}{6}+2k\pi\,\,(k\in Z)$ és $x=\frac{5\pi}{6}+2l\pi\,\,(l\in Z)$ esetén lesz a legnagyobb (ez az érték 1/2).

Statistics:

345 students sent a solution.

5 points: 185 students.

4 points: 91 students.

3 points: 12 students.

2 points: 6 students.

1 point: 5 students.

0 point: 44 students.

Unfair, not evaluated: 2 solutionss.

Problems in Mathematics of KöMaL, November 2006

345 students sent a solution.
5 points:	185 students.
4 points:	91 students.
3 points:	12 students.
2 points:	6 students.
1 point:	5 students.
0 point:	44 students.
Unfair, not evaluated:	2 solutionss.

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