I have the following inequality:
$$ \sqrt{\frac{a}{2x}}+ \frac{b}{1-\frac{a}{a+\sqrt{a(2bx + a)}}} + \frac{a}{2x\frac{a}{a+\sqrt{a(2bx + a)}}\left(1-\frac{a}{a+\sqrt{a(2bx + a)}}\right)} < f $$ on the domain $a>\ln(2)$, $b > 0$, $0 < f < 1$, $x > 1$. Let's call the l.h.s. the "quantity of interest". I want to solve it for $x$.
I use Reduce:
Reduce[{Sqrt[a/(2 x)] +
b/(1 - a/(a + Sqrt[a (2 x b + a)])) +
a / (2 x (a/(a + Sqrt[a (2 x b + a)])) (1 - a/(a + Sqrt[a (2 x b + a)]))) < f,
a > Log[2], 0 < f < 1, 0 <= b, x >= 1}, x]
And get:
0 < f < 1 && 0 <= b < f && a > Log[2] &&
x > Root[-8 a^3 + (a^2 - 16 a^2 b + 8 a^2 f +
16 a^2 f^2) #1 + (-4 a b^2 + 8 a b f - 16 a b^2 f - 4 a f^2 +
32 a b f^2 - 16 a f^3) #1^2 + (4 b^4 - 16 b^3 f +
24 b^2 f^2 - 16 b f^3 + 4 f^4) #1^3 &, 3]
Let's call the Root[....] expression the "implicit root expression". Let's now find an explicit expression for this root (which one can check is real)
FullSimplify[
ToRadicals[
Root[-8 a^3 + (a^2 - 16 a^2 b + 8 a^2 f +
16 a^2 f^2) #1 + (-4 a b^2 + 8 a b f - 16 a b^2 f - 4 a f^2 +
32 a b f^2 - 16 a f^3) #1^2 + (4 b^4 - 16 b^3 f +
24 b^2 f^2 - 16 b f^3 + 4 f^4) #1^3 &, 3]]]
We get:
(1/(96 (b - f)^4))(32 a (b - f)^2 (1 + 4 f) + (8 I (I + Sqrt[3]) a^2 (b - f)^4
(48 b + (1 + 4 f)^2))/(12 Sqrt[3] Sqrt[a^6 (b - f)^12 (-1 + 2 b + 2 f)^2
(27 b^2 - f^2 (1 + 16 f) - b (1 + 18 f))] + a^3 (b - f)^6 (-1 - 12 f +
8 (27 b^2 + (21 - 8 f) f^2 + 18 b (1 + f))))^(1/3) - 8 (1 + I Sqrt[3])
(12 Sqrt[3] Sqrt[a^6 (b - f)^12 (-1 + 2 b + 2 f)^2 (27 b^2 - f^2 (1 + 16 f) -
b (1 + 18 f))] + a^3 (b - f)^6 (-1 - 12 f + 8 (27 b^2 + (21 - 8 f) f^2
+ 18 b (1 + f))))^(1/3))
Let's call this expression the "explicit root expression".
One could simplify by hand further, but let's not do it now (I did it by hand, and then also tried with the simpler expression, but the result doesn't change).
Assume now that we somehow got some specific values for a, b, and f and we want to find x. Specifically: a = Log[20], b = 0.00672874, and f = 0.03.
If I replace these values in the explicit root expression, we get:
2144.31 - 1.47874*10^-12 I
Ok, let's assume that the very small imaginary part is actually 0, as it should be. We have then that, according to the result of the first Reduce, if we have x > 2144.31, then the quantity of interest should be $<f$. But if we plug 2145 and the fixed values for a, b, and f in the quantity of interest, Mathematica returns:
0.0391055
which is not < 0.03.
Let's do a different check then, and plug the values for a,b, and f in the implicit root expression. Mathematica returns
4030.04
and indeed, if we evaluate the quantity of interest for 4031, we get
0.0299971
which is smaller than 0.03, as requested.
Am I doing something wrong when calling ToRadicals or FullSimplify or something else? Is it a precision problem?
ToRadicals
cannot make a choice consistent withRoot
in the presence of parameters (that is to say, for some values of parameters the result might correspond to a differentRoot
solution for the same equation). This is discussed under "Possible Issues" on the refguide page forToRadicals
. $\endgroup$