# 2006 Cyprus MO/Lyceum/Problem 4

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## Problem

Given the function $f(x)=\alpha x^2 +9x+ \frac{81}{4\alpha}$ , $\alpha \neq 0$ Which of the following is correct, about the graph of $f$? $\mathrm{(A)}\ \text{intersects x-axis}\qquad\mathrm{(B)}\ \text{touches y-axis}\qquad\mathrm{(C)}\ \text{touches x-axis}\qquad\mathrm{(D)}\ \text{has minimum point}\qquad\mathrm{(E)}\ \text{has maximum point}$

## Solution $$\alpha x^2+9x+\frac{81}{4\alpha}=\left(\sqrt{\alpha}x+\frac{9}{2\sqrt{\alpha}}\right)^2$$ Notice that if $f(x) = 0$, then $x$ has the unique root of $-\frac{\frac{9}{2\sqrt{\alpha}}}{\sqrt{\alpha}} = \frac{-9}{2\alpha}$, so it touches the x-axis, $\mathrm{(C)}$.

From above, $\mathrm{(A)}$ is not correct because the graph does not intersect the x-axis (it is tangent to it). $\mathrm{(B)}$ is not true; the graph intersects the y-axis since the parabola opens up or down. $\mathrm{(D)}$ and $\mathrm{(E)}$ depend upon the value of $\alpha$; if $\alpha > 0$, then the parabola has a minimum, and if $\alpha > 0$ then the parabola has a maximum.

Thus, the answer is $\mathrm{(C)}$.

## See also

 2006 Cyprus MO, Lyceum (Problems) Preceded byProblem 3 Followed byProblem 5 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
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