Question

Find the value of p that maximizes p(A-bp) - c(A-bp)

Answer 1

To maximize, set f'(p)=0 where,
$$
f(p)=p(A-bp) - c(A-bp)\\
f'(p)=A+b (c-2 p)=0\\
\implies p_{max}= (A+b c)/(2 b)
$$
To show this is a maximum, see if f''(p_{max}) < 0
$$
f''(p)= -2 b
$$
So f''(p) is maximum iff b > 0.