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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?

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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).

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