Problem:
Find the largest positive value attained by the function
f(x)=8x−x2−14x−x2−48,x a real number.
Answer Choices:
A. 7−1
B. 3
C. 23
D. 4
E. 55−5
Solution:
Completing the squares, we have
f(x)=16−(x−4)2−1−(x−7)2
The first term is the formula for the y-coordinate of the upper half of the circle with center at (4,0) and radius 4, and the second term is the formula for the y-coordinate of the upper half-circle with center at (7,0) and radius 1. By examining the graphs of these two semicircles, it is clear that f(x) is real-valued only when 6≤x≤8, and that the maximum value will be attained when x=6. Evaluating, we get
f(6)=16−(6−4)2=12=23
OR
Note that f(x)=(8−x)x−(8−x)(x−6) is a real number if and only if 6≤x≤8. Note that
For all x such that 6≤x≤8, the numerator of this last expression is maximized and its denominator is minimized when x=6. Hence, its maximum is f(6)=662=23.
