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

In summary, the proof shows that if ab < 0, then the equation ax^3 + bx + c = 0 has at most three real roots. This is because when the absolute value of 3ax^2 is equal to the absolute value of b, the equation f'(x) = 0 has only two roots. Therefore, the given equation can have at most three real roots when ab < 0. This conclusion is also supported by the fact that all cubic polynomial equations have at most three real roots.
  • #1
Joe20
53
1
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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  • #2
But all cubic polynomial equations have at most three real roots! (Nerd)
 
  • #3
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?
 

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

1. How is Rolle's theorem used to prove for roots?

Rolle's theorem is a mathematical theorem that states that if a function is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), with f(a) = f(b), then there exists at least one point c in the open interval (a,b) where the derivative of the function is equal to zero. This point c is known as a "critical point" and can be used to prove the existence of roots for the function within the interval.

2. Can Rolle's theorem be applied to all types of functions?

Yes, Rolle's theorem can be applied to any continuous and differentiable function on a closed interval. However, it is important to note that the function must also have the same value at the endpoints of the interval in order for the theorem to hold true.

3. How is Rolle's theorem different from the Mean Value Theorem?

Rolle's theorem is a special case of the Mean Value Theorem. While both theorems state that there exists a point where the derivative of a function is equal to zero, Rolle's theorem only applies to functions with equal values at the endpoints of the interval, while the Mean Value Theorem applies to any continuous function on a closed interval.

4. Can Rolle's theorem be used to find all roots of a function?

No, Rolle's theorem can only be used to prove the existence of at least one root within a given interval. It does not provide any information about the location or number of roots of a function.

5. How can I determine the critical points of a function using Rolle's theorem?

To determine the critical points of a function using Rolle's theorem, you must first find the derivatives of the function and then set them equal to zero. The values of x that result from this equation will be the critical points, which can then be used to prove the existence of roots within the given interval.

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