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akhil samir killawala Grade: 11
        

f(x)=x3+px+q


find the condition on p and q so that f(x) has 3 distinct real roots..


please reply as early as possible

7 years ago

Answers : (1)

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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