Dear friends - I have a quotient which I need to differentiate and show the
differential is always positive for unknown H>0. We have 7 unknown
constants all known to be positive and less than 1 except for A which may
be any positive value. In Sage Cell Server I did this
var('A','K1','K2','K3','f1','f2','f3','H'); B =
-A*(K1*f1*f2*f3*H^2+2*K1*K2*f1*f3*H+3*K1*K2*K3*f2);D=f1^3*f2*f3*H^3+K1*f1*f2*f3*H^2+K1*K2*f1*f3*H+K1*K2*K3*f2;diff(B,H)*D-diff(D,H)*B
where B is the numerator and D is the denominator of the quotient.
The result is
(3*H^2*f1^3*f2*f3 + 2*H*K1*f1*f2*f3 + K1*K2*f1*f3)*(H^2*K1*f1*f2*f3 +
2*H*K1*K2*f1*f3 + 3*K1*K2*K3*f2)*A -
2*(H^3*f1^3*f2*f3 + H^2*K1*f1*f2*f3 + H*K1*K2*f1*f3 +
K1*K2*K3*f2)*(H*K1*f1*f2*f3 + K1*K2*f1*f3)*A
which I believe I can see must be positive when the K's and f's are positive
but less than 1
Could I reach this conclusion in a more definite way?
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