# Mathematica Inequality with Assumptions

Is there a reason why Mathematica cannot verify the following?

vv = t (a^2 + b^2)/2  - (a - b) (a + b) (1 - Exp[- 2 c^2 b^2 t])/( 4 c^2 b^2)
FullSimplify[
Positive[vv],
Assumptions -> Flatten[
{
Thread[{a, b, t, c} > 0],
Element[{a, b, t, c}, Reals],
a > b
}
]
]


Which just gives me

Positive[-(((a - b) (a + b) (1 - E^(-2 b^2 c^2 t)))/(4 b^2 c^2)) + 1/2 (a^2 + b^2) t]


However, I'm fairly certain it's true, since it is shown here.

• From the help on Positive: "Quantities that are not NumericQ remain unevaluated". vv is symbolic Jun 27 '17 at 17:06
• @Raffaele Ah, thanks. How would I get it to show that it must be positive then? Jun 27 '17 at 17:45

$Assumptions = {0 < b < a, c > 0, t > 0}; T[a_, b_, c_, t_] := (a^2 + b^2) t/2 - (a^2 - b^2) (1 - Exp[-2 b^2 c^2 t])/(4 b^2 c^2); Reduce[T[a, b, c, 0] == 0 && D[T[a, b, c, t], t] > 0, Reals] // Simplify  I get: True which shows that$T > 0 \; \; \forall\; 0 < b < a \, \land \, c > 0\, \land \, t>0\$.