1993 AHSME Problems/Problem 26: Difference between revisions
| Line 1: | Line 1: | ||
== Problem == | == Problem == | ||
Find the largest positive value attained by the function | Find the largest positive value attained by the function | ||
<math>f(x)=\sqrt{8x-x^2}-\sqrt{14x-x^2-48}</math>, x a real number. | <math>f(x)=\sqrt{8x-x^2}-\sqrt{14x-x^2-48}</math>, <math></math>x a real number. | ||
Revision as of 11:25, 27 November 2020
Problem
Find the largest positive value attained by the function
, $$ (Error compiling LaTeX. Unknown error_msg)x a real number.
Solution
We can rewrite the function as 
and then factor it to get 
. From the expressions under the square roots, it is clear that 
is only defined on the interval ![$[6, 8]$](http://latex.artofproblemsolving.com/c/a/c/cac44bad2a2c9ce7acd302172e2beda986d44564.png)
.
The 
factor is decreasing on the interval. The behavior of the 
factor is not immediately clear. But rationalizing the numerator, we find that 
, which is monotonically decreasing. Since both factors are always positive, is also positive. Therefore, is decreasing on , and the maximum value occurs at 
. Plugging in, we find that the maximum value is 
.
See also
| 1993 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 25 |
Followed by Problem 27 | |
| 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 | ||
| All AHSME Problems and Solutions | ||
These problems are copyrighted © by the Mathematical Association of America. 
