Difference between revisions of "1983 AIME Problems/Problem 9"
Aimesolver (talk | contribs) m (→Solution 2) |
Aimesolver (talk | contribs) m (→Solution 2) |
||
Line 21: | Line 21: | ||
<math>f'(y)</math> = <math>9 - 4y^{-2}</math> | <math>f'(y)</math> = <math>9 - 4y^{-2}</math> | ||
− | <math>f'(y)</math> is zero only when <math>y = \frac{2}{3}</math> or <math>y = -\frac{2}{3}</math>. It can further be verified that <math>\frac{2}{3}</math> and <math>-\frac{2}{3} are relative minima by finding the derivatives of other points near the critical points. However, since < | + | <math>f'(y)</math> is zero only when <math>y = \frac{2}{3}</math> or <math>y = -\frac{2}{3}</math>. It can further be verified that <math>\frac{2}{3}</math> and <math>-\frac{2}{3}</math> are relative minima by finding the derivatives of other points near the critical points. However, since <math>x \sin{x}</math> is always positive in the given domain, <math>y = \frac{2}{3}</math>. Therefore, <math>x\sin{x}</math> = <math>\frac{2}{3}</math>, and the answer is <math>\frac{(9)(\frac{2}{3})^2 + 4}{\frac{2}{3}} = \boxed{012}</math>. |
== See also == | == See also == |
Revision as of 21:03, 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
. It can further be verified that
and
are relative minima by finding the derivatives of other points near the critical points. 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 |