A Real Equation!
If f ( x ) = a 1 x 3 + a 2 x 2 + a 3 = 0 where a 1 , a 2 ∈ R + and a 3 ∈ R has three distinct real roots, then the exhaustive range of a 3 is a 3 ∈ ( − d a 1 e b a 2 c , − f ) where b , c , d , e , f ∈ I and g cd ( b , d ) = 1 , enter answer as b + c + d + e + f .
All of my problems are original
The answer is 36.
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1 solution
@Aryan Sanghi a very nice and interesting problem . Thanks. Upvoted.
Or you could use cubic discriminant by setting c = 0 , then solve for the quadratic inequality of d .
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That's an interesting method. Thanku for sharing. :)
@Aryan Sanghi what is the meaning of exhaustive range?
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It means all values possible of c are covered in the range. :)
Now,
f ( x ) = a 1 x 3 + a 2 x 2 + a 3 f ′ ( x ) = 3 a 1 x 2 + 2 a 2 x
For maxima and minima, f ′ ( x ) = 0 , i.e. 3 a 1 x 2 + 2 a 2 x = 0 or x = 0 = k 1 and x = − 3 a 1 2 a 2 = k 2
Now, if f ( x ) = 0 has three distinct real roots, then maxima and minima are of opposite signs or f ( k 1 ) × f ( k 2 ) < 0
f ( 0 ) × f ( − 3 a 1 2 a 2 ) < 0 ( a 1 ( 0 ) 3 + a 2 ( 0 ) 2 + a 3 ) ( a 1 ( − 3 a 1 2 a 2 ) 3 + a 2 ( − 3 a 1 2 a 2 ) 2 + a 3 ) < 0
a 3 ( a 3 + 2 7 a 1 2 4 a 2 3 ) < 0
a 3 ∈ ( − 2 7 a 1 2 4 a 2 3 , 0 )
Therefore, b = 4 , c = 3 , d = 2 7 , e = 2 , f = 0 , b + c + d + e + f = 3 6