1971 Canadian MO Problems/Problem 2
Contents
[hide]Problem
Let and
be positive real numbers such that
. Show that
.
Solution
Solution 1
which is true since by AM-GM, we get:
and we are given that
, so
Solution 2
Let . Since we want to find the minimum of the function over the interval
, we can take the derivative. Using the product rule, we get
. Since we want this value to be zero, the numerator must be zero. But the only root of the equation
is
, and so plugging the answer back in we have
Thus the minimum is
and we are done.
See Also
1971 Canadian MO (Problems) | ||
Preceded by Problem 1 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • | Followed by Problem 3 |