I'm trying to figure out whether an expression is always positive given positive parameters. When the expression is complicated, I can't do this by eye. Is there any way to make Mathematica prove (that is, check) if j
is positive for positive c, d, k, ton, toff, V
?
j=-2 c d k toff ton - d^2 k toff ton - 2 c k toff^2 ton -
2 d k toff^2 ton - k toff^3 ton - 2 c d k ton^2 - d^2 k ton^2 -
4 c k toff ton^2 - 4 d k toff ton^2 - 3 k toff^2 ton^2 -
2 c k ton^3 - 2 d k ton^3 - 3 k toff ton^3 -
k ton^4 + (Sqrt[k] Sqrt[ton] (toff + ton) Sqrt[
d + toff +
ton] (2 c + d + toff +
ton) \[Sqrt]((c + d) k ton (c + toff + ton) (d + toff + ton) +
4 c d ((d + toff + ton) (k toff + (toff + ton)^2) +
c (k toff +
d (toff + ton) + (toff + ton)^2)) V))/(2 Sqrt[(c +
d) (c + toff + ton)]) + (Sqrt[k] Sqrt[
ton] (toff + ton) Sqrt[(c + d) (c + toff + ton)] Sqrt[
d + toff +
ton] (k ton (d + toff + ton) (2 c + d + toff + ton) +
4 d ((d + toff + ton) (k toff + (toff + ton)^2) +
2 c (k toff +
d (toff + ton) + (toff + ton)^2)) V))/(2 \[Sqrt]((c +
d) k ton (c + toff + ton) (d + toff + ton) +
4 c d ((d + toff + ton) (k toff + (toff + ton)^2) +
c (k toff + d (toff + ton) + (toff + ton)^2)) V))
I have many more expressions like this. I have tried putting in random values of the parameters and I do (so far) always get positive values. But I'd prefer something more convincing than a Monte Carlo argument for this (and other) complicated expressions being positive for any positive inputs. Thoughts?