Issues with an application of definite internal and marginal utilities

Question

I'm going to state the problem without much context because it is an issue of the math I'm doing, rather than any specific econ concept.

Condition. I'm trying to prove this.
$$ V(q_l) - \theta_l*q_l > V(q_h) - \theta_h*q_h$$

Given $$q_l>q_h$$ $$\theta_l < \theta_h $$ $$V'() > 0$$ $$ V'' < 0 $$ $$V'(q_l) = \theta_l $$ $$V'(q_h) = \theta_h $$

I'm trying to use definite integrals to prove this. But I think I'm getting something fundamentally wrong here.

$$ \int_0^{q_i} V'(q)dq = \int_0^{q_i} \theta_i $$ $$ V(q_i) - V(0) = \theta_i*q_i$$

This gives two equations, one for each $q_l, q_h$. Substracting these two equations from one another.

$$ V(q_l) - V(q_h) = \theta_l*q_l -\theta_h*q_h $$

I'm definitely doing something wrong here. Any help?

Answer 1

The mistake you are making appears to be treating $\theta_i$ as a variable while in reality it is a specific value of the derivative.

So $$\int_0^{q_i} V'(q)dq = \int_0^{q_i} \theta_i dq \;\;\; i=l,h$$

is wrong, since $V'(q)$ equals $\theta_l$ evaluated at $q_l$ only (and equals $\theta_h$ evaluated at $q_h$ only).