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```        f(x)=x3+px+q
find the condition on p and q so that f(x) has 3 distinct real roots..
7 years ago

Ramesh V
70 Points
```										Suppose  that f(x) has three distinct roots. Then are are x1 and x2 (by Rolle's  theorem) sitting between these roots such that f'(x1) = f'(x2) = 0:
Since  f'(x) is a quadratic with roots x1 and x2; it follows that p < 0:  Setting f0(x) = 0
we get  x1 := -(p/3)1/2 and x2 =  -(p/3)1/2 From the second derivative we see that f has a local maximum and a local  minimum at x1 and x2; respectively. Therefore f(x1) > 0 and f(x2)  < 0; that is,
f(x1)f(x2) < 0        which gives 27q2  + 4p3 < 0
Conversely, suppose that 27q2 + 4p3 < 0: Then obviously p < 0:  Therefore f'(x) = 0 has two roots x1 and x2 at which f has a local max  and a local min. That there is a root between x1 and x2 follows from the  fact that f(x1)f(x2) < 0
(so apply IVT). Since f(x)  --> - infinity as x --> - infinity and  f(x) -->  infinity as x  --> infinity; it follows that f has one root in ( - infinity, x1) and another in (x2, infinity):
Hence f  has three roots.
--
regards
Ramesh
```
7 years ago
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