Vile Villainous Vivacious Varying Vaulting Vieta Values
⎩ ⎪ ⎨ ⎪ ⎧ a + b + c + d a b + a c + a d + b c + b d + c d a b c + a b d + a c d + b c d = 1 2 = 4 6 = 6 0 Given that a < b < c < d , what is the supremum of the product a b c d ?
Bonus: What about the infimum?
The answer is 25.
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2 solutions
Voilà! A venerated veteran vouchsafed a valorous veracious veridicous answer. Veritably, a vivacious view.
f(1)=f(5)=u−25<0<u−9=f(3) how did you do that?
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Given that f ( x ) = x 4 − 1 2 x 3 + 4 6 x 2 − 6 0 x + u , then f ( 1 ) = u − 2 5 , f ( 5 ) = u − 2 5 , f ( 3 ) = u − 9 .
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how did you get that relationship u−25<0<u−9 ?
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@Nazmul Hasan Shipon – A quartic with leading coefficient 1 and three turning points at 1 , 3 , 5 is going to have a graph looking like a smooth W. To have four real roots, this W has to cross the x -axis four times, and so the left and right minima of the W must be below the x -axis and the central maximum must be above the x -axis. Thus we need f ( 1 ) , f ( 5 ) < 0 < f ( 3 ) .
Verily, Vieta vanquishes: ( a , b , c , d ) are the roots of x 4 − 1 2 x 3 + 4 6 x 2 − 6 0 x + P = 0
where P = a b c d . This needs to have four real roots; the range of P over which this is possible is the range we're after. One option is to use the discriminant of the quartic.
Alternatively, put u = x − 3 , so that the equation becomes u 4 − 8 u 2 − 9 + P = 0
We can treat this as a quadratic in u 2 ; solving gives u 2 = 4 ± 2 5 − P
For this to have four distinct real roots, we need P < 2 5 and 4 − 2 5 − P > 0
which works out to 9 < P . Hence the infimum of P is 9 and the supremum is 2 5 .
Now a , b , c , d are the four distinct real roots of the quartic f ( x ) = x 4 − 1 2 x 3 + 4 6 x 2 − 6 0 x + u where u = a b c d , and we note that f ′ ( x ) = 4 x 3 − 3 6 x 2 + 9 2 x − 6 0 = 4 ( x 3 − 9 x 2 + 2 3 x − 1 5 ) = 4 ( x − 1 ) ( x − 3 ) ( x − 5 ) Thus we must have f ( 1 ) = f ( 5 ) = u − 2 5 < 0 < u − 9 = f ( 3 ) if there are to be four real roots. Thus we deduce that 9 < u < 2 5 . so the supremum of u = a b c d is 2 5 , while the infimum is 9 .