# Question:How to "show" this inequality?

## Question:How to "show" this inequality?

Maple 2024

I have two expressions, wo_theta and with_theta, which depend on multiple variables.

I would need your help to:

1. Verify, as formally as possible, that wo_theta > with_theta always, i.e., for any value of theta different from zero (and regardless of the values taken up by the other variables)
2. Show the above in a way that is easy and immediate to interpret (perhaps using some type of plot?)

In other words, I want to verify that as soon as I introduce any theta in my expression such expression becomes smaller:

 > restart;
 > local gamma;
 (1)
 > assume(0 < gamma, 0 < nu__02, 0 < nu__01, 0 <= sigma__v, delta__1::real, delta__2::real, delta__3::real, theta::real); interface(showassumed=0);
 (2)
 > wo_theta := X__3*(-X__3*lambda__3 - delta__3*lambda__3 + DEV) + X__2*(-X__2*lambda__2 - delta__2*lambda__2 - nu__02) + X__1*(-X__1*lambda__1 - delta__1*lambda__1 - nu__01) + X__2*(nu__02 + DEV/2) + X__1*(nu__01 + DEV/2) - gamma*X__2^2*sigma__v^2/4 - gamma*X__1^2*sigma__v^2/4 + gamma*X__2*X__1*sigma__v^2/2;
 (3)
 > with_theta := X__3*(-X__3*lambda__3 - theta*lambda__3 - delta__3*lambda__3 + DEV) + X__2*(-X__2*lambda__2 + theta*lambda__2 - delta__2*lambda__2 - nu__02) + X__1*(-X__1*lambda__1 + theta*lambda__1 - delta__1*lambda__1 - nu__01) + X__2*(nu__02 + DEV/2) + X__1*(nu__01 + DEV/2) - gamma*X__2^2*sigma__v^2/4 - gamma*X__1^2*sigma__v^2/4 + gamma*X__2*X__1*sigma__v^2/2 + theta*(lambda__1*(X__1 + delta__1 - theta) + lambda__2*(X__2 + delta__2 - theta) - lambda__3*(X__3 + delta__3 + theta));
 (4)
 > collect(with_theta, theta);
 (5)
 > solve(wo_theta > with_theta, theta) assuming 0 < gamma, 0 < nu__02, 0 < nu__01, 0 < sigma__v, delta__1::real, delta__2::real, delta__3::real, theta::real;
 > solve(with_theta < wo_theta, theta);
 > difference_term := (-lambda__1 - lambda__2 - lambda__3)*theta^2 + (X__1*lambda__1 + X__2*lambda__2 - X__3*lambda__3 + lambda__1*(X__1 + delta__1) + lambda__2*(X__2 + delta__2) - lambda__3*(X__3 + delta__3))*theta;
 (6)
 > # I would expect such difference_term in theta to be always < 0, i.e., for any theta different from 0) # (Note that lambda_1, lambda_2, and lambda_3 are always > 0, while theta, the three X and the three delta can be positive or negative. In other words, it suffices to show that the linear term in theta is always negative...) solve(difference_term<0);
 (7)