Difference between revisions of "1983 AIME Problems/Problem 9"
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== Solution 2 == | == Solution 2 == | ||
− | Let <math>y = x\sin{x}</math> and rewrite the expression as <math>f(y) = 9y + \frac{4}{y}</math>. To minimize <math>f(y)</math>, take the derivative of <math>f(y)</math> and set it equal to zero. The derivative of <math>f(y)</math>, using the [[Power Rule]] is | + | Let <math>y = x\sin{x}</math> and rewrite the expression as <math>f(y) = 9y + \frac{4}{y}</math>, similar to the previous solution. To minimize <math>f(y)</math>, take the [[derivative]] of <math>f(y)</math> and set it equal to zero. The derivative of <math>f(y)</math>, using the [[Power Rule]] is |
<math>f'(y)</math> = <math>9 - 4y^{-2}</math> | <math>f'(y)</math> = <math>9 - 4y^{-2}</math> |
Revision as of 20:28, 5 February 2010
Contents
Problem
Find the minimum value of for
.
Solution
Let . We can rewrite the expression as
.
Since and
because
, we have
. So we can apply AM-GM:
The equality holds when .
Therefore, the minimum value is (when
; since
is continuous and increasing on the interval
and its range on that interval is from
, by the Intermediate Value Theorem this value is attainable).
Solution 2
Let and rewrite the expression as
, similar to the previous solution. To minimize
, take the derivative of
and set it equal to zero. The derivative of
, using the Power Rule is
=
is zero only when
or
. However, since
is always positive in the given domain,
. Therefore,
=
, and the answer is
.
See also
1983 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |