P(x) =
2
24
500
5
100
x −x
−
or
P′(x) = 24
5
x50
−
Now P′(x) = 0 gives x = 240 Also 
1
P ( )
50
x
−
′′
= So 
1
P (240)
0
−50
′′
=
<
Thus, x = 240 is a point of maxima Hence, the manufacturer can earn maximum
profit, if he sells 240 items