Let

$$ S = 4 ( alpha -2) beta -58 ( alpha -2) ^ 2- beta ^ 2-8 c_6 left (7 alpha +2 c_6-13 right) $$

$$ T = – frac { left (( alpha -4) ( alpha -2) ( alpha (3 alpha +4 beta -27) -18 beta +60) +4 c_2 ( alpha + beta -5) right) {} ^ 2} {4 ( alpha + beta -5) ^ 2} + c_2 ( alpha + beta -5) left (8 alpha + beta +4 c_6-16 right) {} ^ 2-c_2 ( alpha + beta -5) left (2 ( alpha -2) (3 alpha +10 beta -6) +8 c_6 ( alpha + beta -3) right) $$

$$ U = left (- frac {1} {12} left (-12 c_4 ( alpha + beta -4) -36 c_2 right) left (-2 alpha + beta -2 left (-5 alpha -2 c_6 + 10 right) +4 right) {} ^ 2- frac {1} {6} c_6 left ( frac {3 ( alpha -6) ( alpha – 5) ( alpha -4) ( alpha -2)} { alpha + beta -5} -6 ( alpha -4) (2 alpha -9) ( alpha -2) -12 c_2 right ) left (-2 alpha + beta -2 left (-5 alpha -2 c_6 + 10 right) +4 right) – frac {1} {6} left ( frac {3 ( alpha -6) ( alpha -5) ( alpha -4) ( alpha -2)} { alpha + beta -5} -6 ( alpha -4) (2 alpha -9) ( alpha -2) -12 c_2 right) left (-3 ( alpha -2) (3 alpha -11) -2 c_6 ( alpha + beta -3) -6 c_4 right) + frac { 1} {12} left (-12 c_4 ( alpha + beta -4) -36 c_2 right) left (-2 ( alpha -2) (7 alpha -24) -4 ( alpha + beta -3) left (-5 alpha -2 c_6 + 10 right) right) right) {} ^ 2-4 left (- frac {1} {36} left ( frac { 3 ( alpha -6) ( alpha -5) ( alpha -4) ( alpha -2)} { alpha + beta -5} -6 ( alpha -4) (2 alpha -9) ( alpha -2) -12 c_2 right) {} ^ 2 + c_2 ( alpha + beta -5) left (-2 alpha + beta -2 left (-5 alpha -2 c_6 + 10 right) +4 right t) {} ^ 2-c_2 ( alpha + beta -5) left (-2 ( alpha -2) (7 alpha -24) -4 ( alpha + beta -3) left (- 5 alpha -2 c_6 + 10 right) right) right) left (- frac {1} {4} c_6 ^ 2 left (-2 ( alpha -2) (7 alpha -24) -4 ( alpha + beta -3) left (-5 alpha -2 c_6 + 10 right) right) – frac {1} {2} c_6 left (-2 alpha + beta – 2 gauche (-5 alpha -2 c_6 + 10 right) +4 right) left (-3 ( alpha -2) (3 alpha -11) -2 c_6 ( alpha + beta -3) -6 c_4 right) – frac {1 } {4} left (-3 ( alpha -2) (3 alpha -11) -2 c_6 ( alpha + beta -3) -6 c_4 right) {} ^ 2 + c_4 left (- 2 alpha + beta -2 left (-5 alpha -2 c_6 + 10 right) +4 right) {} ^ 2-c_4 left (- 2 ( alpha -2) (7 alpha – 24) -4 ( a lpha + beta -3) left (-5 alpha -2 c_6 + 10 right) right) right) $$,

or $ c_2, c_4, c_6 in mathbb {R} $ and $ alpha, beta> 0 $. My question is the following:

Are there $ alpha, beta> 0 $ and $ c_2, c_4, c_6 in mathbb {R} $ such as $ S> 0 $, $ T> 0 $, $ U le0 $?

I used the codes

```
FindInstance(S>0 && T>0 && U<=0 && α > 0 && β >0, {c2,c4,c6,α,β}, Reals)
```

But it lasts more than 5 minutes and no results can be obtained. So I pick up several α on $ (1 / 10.10) $ like that

```
α=17/7; FindInstance(S>0 && T>0 && U<=0 && α > 0 && β >0, {c2,c4,c6,β}, Reals)
```

He replied with an empty set

```
{}
```

It seems that the interval of $ alpha $ is so narrow and hard to find. Any other approach to get a single solution $ ( alpha, beta, c_2, c_4, c_6) $?

Any reference, suggestion, idea or comment is welcome. Thank you!

```
S=-58 (-2 + α)^2 + 4 (-2 + α) β - β^2 - 8 Subscript(c, 6) (-13 + 7 α + 2 Subscript(c, 6))
T=-((-4 + α) (-2 + α) (60 -
18 β + α (-27 + 3 α + 4 β)) +
4 (-5 + α + β) Subscript(c,
2))^2/(4 (-5 + α + β)^2) + (-5 + α + β) Subscript(c, 2) (-16 + 8 α + β + 4 Subscript(c, 6))^2 - (-5 + α + β) Subscript(c, 2) (2 (-2 + α) (-6 + 3 α + 10 β) + 8 (-3 + α + β) Subscript(c, 6))
U=(-(1/12) (-36 Subscript(c, 2) -
12 (-4 + α + β) Subscript(c, 4)) (4 -
2 α + β -
2 (10 - 5 α - 2 Subscript(c, 6)))^2 + 1/12 (-36 Subscript(c, 2) -
12 (-4 + α + β) Subscript(c,
4)) (-2 (-2 + α) (-24 + 7 α) -
4 (-3 + α + β) (10 - 5 α -
2 Subscript(c, 6))) - 1/6 (-6 (-4 + α) (-2 + α) (-9 + 2 α) + (
3 (-6 + α) (-5 + α) (-4 + α) (-2 + α))/(-5 + α + β) - 12 Subscript(c, 2)) (4 - 2 α + β -
2 (10 - 5 α - 2 Subscript(c, 6))) Subscript(c, 6) - 1/6 (-6 (-4 + α) (-2 + α) (-9 + 2 α) + (
3 (-6 + α) (-5 + α) (-4 + α) (-2 + α))/(-5 + α + β) - 12 Subscript(c, 2)) (-3 (-2 + α) (-11 + 3 α) - 6 Subscript(c, 4) - 2 (-3 + α + β) Subscript(c, 6)))^2 - 4 (-(1/36) (-6 (-4 + α) (-2 + α) (-9 + 2 α) + (
3 (-6 + α) (-5 + α) (-4 + α) (-2 + α))/(-5 + α + β) - 12 Subscript(c, 2))^2 + (-5 + α + β) Subscript(c, 2) (4 - 2 α + β - 2 (10 - 5 α - 2 Subscript(c, 6)))^2 - (-5 + α + β) Subscript(c, 2) (-2 (-2 + α) (-24 + 7 α) - 4 (-3 + α + β) (10 - 5 α -
2 Subscript(c, 6)))) (Subscript(c, 4) (4 - 2 α + β -
2 (10 - 5 α - 2 Subscript(c, 6)))^2 -
Subscript(c, 4) (-2 (-2 + α) (-24 + 7 α) - 4 (-3 + α + β) (10 - 5 α - 2 Subscript(c, 6))) -
1/4 (-2 (-2 + α) (-24 + 7 α) -
4 (-3 + α + β) (10 - 5 α -
2 Subscript(c, 6))) !(*SubsuperscriptBox((c), (6), (2))) -
1/2 (4 - 2 α + β -
2 (10 - 5 α - 2 Subscript(c, 6))) Subscript(c,
6) (-3 (-2 + α) (-11 + 3 α) - 6 Subscript(c, 4) -
2 (-3 + α + β) Subscript(c, 6)) -
1/4 (-3 (-2 + α) (-11 + 3 α) - 6 Subscript(c, 4) -
2 (-3 + α + β) Subscript(c, 6))^2)
```