I was wondering if anyone can help me solve this problem.

P(t) = Max{t x W(x) - C(x)}

P(t)max occurs at x*(t)

W(x) is a strictly increasing function.

t falls between 0,1

What is the relationship between x*(.) and t?

i got to

dp/dx = t x W'(x) - C'(x) = 0 at x*(t)

from that i deduced dt/dx*(t) = (C'' x W' - W'' x C')/(W')^2 but i know nothing about the signs of the second derivatives.