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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.

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2
  • $\begingroup$ From the help on Positive: "Quantities that are not NumericQ remain unevaluated". vv is symbolic $\endgroup$
    – Raffaele
    Jun 27, 2017 at 17:06
  • $\begingroup$ @Raffaele Ah, thanks. How would I get it to show that it must be positive then? $\endgroup$ Jun 27, 2017 at 17:45

1 Answer 1

2
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Writing:

$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 $T > 0 \; \; \forall\; 0 < b < a \, \land \, c > 0\, \land \, t>0$.

Other effective strategies don't come to my mind.

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