Difference between revisions of "2006 Cyprus MO/Lyceum/Problem 4"
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==Problem== | ==Problem== | ||
| − | {{ | + | Given the function <math>f(x)=\alpha x^2 +9x+ \frac{81}{4\alpha}</math> , <math>\alpha \neq 0</math> |
| + | Which of the following is correct, about the graph of <math>f</math>? | ||
| + | |||
| + | A. intersects x-axis | ||
| + | |||
| + | B. touches y-axis | ||
| + | |||
| + | C. touches x-axis | ||
| + | |||
| + | D. has minimum point | ||
| + | |||
| + | E. has maximum point | ||
==Solution== | ==Solution== | ||
| − | {{ | + | <math>\alpha x^2 + 9x + \frac{81}{4\alpha} = \left(\sqrt{\alpha}x + \frac{9}{2\sqrt{\alpha}}\right)^2</math>. Notice that if <math>f(x) = 0</math>, then <math>x</math> has the unique root of <math>-\frac{\frac{9}{2\sqrt{\alpha}}}{\sqrt{\alpha}} = \frac{-9}{2\alpha}</math>, so it touches the x-axis, <math>\mathrm{(C)}</math>. |
| + | |||
| + | From above, <math>\mathrm{(A)}</math> is not correct because the graph does not intersect with the x-axis (it is tangent to it). <math>\mathrm{(B)}</math> is not true; the graph intersects with the y-axis, since the [[parabola]] opens up or down. <math>\mathrm{(D)}</math> and <math>\mathrm{(E)}</math> depend upon the value of <math>\alpha</math>; if <math>\alpha > 0</math>, then the parabola has a minimum, and if <math>\alpha > 0</math> then the parabola has a maximum. | ||
==See also== | ==See also== | ||
{{CYMO box|year=2006|l=Lyceum|num-b=3|num-a=5}} | {{CYMO box|year=2006|l=Lyceum|num-b=3|num-a=5}} | ||
| + | |||
| + | [[Category:Introductory Algebra Problems]] | ||
Revision as of 20:10, 17 October 2007
Problem
Given the function 
, 
Which of the following is correct, about the graph of 
?
A. intersects x-axis
B. touches y-axis
C. touches x-axis
D. has minimum point
E. has maximum point
Solution
. Notice that if 
, then 
has the unique root of 
, so it touches the x-axis, 
.
From above, 
is not correct because the graph does not intersect with the x-axis (it is tangent to it). 
is not true; the graph intersects with the y-axis, since the parabola opens up or down. 
and 
depend upon the value of 
; if 
, then the parabola has a minimum, and if then the parabola has a maximum.
See also
| 2006 Cyprus MO, Lyceum (Problems) | ||
| Preceded by Problem 3 |
Followed by Problem 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 | ||
