MHB Using Rolle's theorem to prove for roots (part 2)

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Hi, I have done up the proof for the question below. Please correct me if I have done wrong for the proof. Thanks in advanced!Question: Prove that if ab < 0 then the equation ax^3 + bx + c = 0 has at most three real roots.Proof:

Let f(x) = ax^3 + bx + c.

Assume that f(x) has 4 distinct roots, f(p) = f(q) = f(r) = f(s) = 0, there is a point x1 \in (p,q) such that f'(x1) = 0 ; x2 \in (q, r) such that f'(x2) = 0 ; x3 \in (r,s) such that f'(x3) = 0.

Since ab < 0 then there are two possibilities where a>0 and b<0 or a <0 , b > 0.

f'(x) = 3ax^2+b

If the absolute value of 3ax^2 = the absolute value of b where 3ax^2 > 0 and b < 0, then f'(x) = 0

If the absolute value of 3ax^2 = the absolute value of b where 3ax^2 < 0 and b > 0, then f'(x) = 0

This is not true because the equation f'(x) = 0 has only two roots.

Hence the given equation has at most three real roots when ab < 0.
 
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But all cubic polynomial equations have at most three real roots! (Nerd)
 
Olinguito said:
But all cubic polynomial equations have at most three real roots! (Nerd)


Here is the revised proof:

Proof:

Let f(x) = ax^3 + bx + c.

Assume that f(x) has 3 distinct roots, f(p) = f(q) = f(r) = 0, there is a point x1 element of (p,q) such that f'(x1) = 0 ; x2 element of (q, r) such that f'(x2) = 0.

Since ab < 0 then there are two possibilities where a>0 and b<0 or a <0 , b > 0.

f'(x) = 3ax^2+b

If the absolute value of 3ax^2 = the absolute value of b where 3ax^2 > 0 and b < 0, then f'(x) = 0

If the absolute value of 3ax^2 = the absolute value of b where 3ax^2 < 0 and b > 0, then f'(x) = 0

f'(x) = 0 has two roots.

Hence the given equation has at most three real roots when ab < 0.

Will this be ok? Or need further improvement? If so, how can it be improved?
 
I posted this question on math-stackexchange but apparently I asked something stupid and I was downvoted. I still don't have an answer to my question so I hope someone in here can help me or at least explain me why I am asking something stupid. I started studying Complex Analysis and came upon the following theorem which is a direct consequence of the Cauchy-Goursat theorem: Let ##f:D\to\mathbb{C}## be an anlytic function over a simply connected region ##D##. If ##a## and ##z## are part of...

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