If the demand function for the DoorKnobs operating system is related to perceived market share s and actual market share x by the equation p  512s(1  x) , then in the long run, the highest price at which DoorKnobs could sustain a market share of 3/4 is

A) $256.
B) $113.78.
C) $96.
D) $128.
E) $81.92.

In the long run, the market share x will converge to the perceived market share s. Therefore, we can rewrite the demand function as p = 512s(1-s).
To sustain a market share of 3/4, DoorKnobs needs to set the price at a point where the demand equals (3/4) times the total market demand. The total market demand is equal to 512s(1-s), so we need to solve the following equation for s:
512s(1-s) = (3/4) * 512
s(1-s) = 3/4
s^2 - s + 3/4 = 0
Solving for s using the quadratic formula, we get:
s = (1 ± sqrt(1 - 4(3/4)))/2
s = (1 ± sqrt(1 - 3))/2
s = (1 ± sqrt(-2))/2
Since we cannot have a negative square root, the only solution is s = (1 + sqrt(-2))/2.
Plugging this value back into the demand function, we get:
p = 512s(1-s) = 512(1 + sqrt(-2)/2)(-sqrt(-2)/2) = 96.
Therefore, the highest price at which DoorKnobs could sustain a market share of 3/4 is $96, which is option C.