# Using Rolle's theorem to prove for roots

• MHB
• Joe20
In summary, if the equation ax^3 + bx + c = 0 has more than one root, then there is a point between the roots at which the derivative is zero.
Joe20
I have deduce a proof as stated below and am not sure if it is correct, therefore need some advice.

Question:
Prove that if ab > 0 then the equation ax^3 + bx + c = 0 has exactly one root by rolle's theoremProof:
Let f(x) = ax^3+bx+c = 0. f(x) is continuous and differentiable since it is a polynomial.

Assume f(x) has 2 roots, f(a) = 0 and f(b) = 0, there is a point d element of (a,b) such that f'(d) = 0.

f'(x) = 3ax^2+b
Since ab>0, a and b must be both positive or both negative.
f'(d) = 3a(d)^2+b = 3ad^2+b not equal to 0 instead >0 since any values of d for d^2 will be positive.

Likewise f'(d) = 3a(d)^2 + b will not equal to 0 instead <0 for all negative values of a and b.

Hence, a contradiction, f has exactly one root.

Re: Using rolle's theorem to prove for roots

Your proof is essentially correct (although in one line you used $a$ and $b$ as roots of the polynomial whereas they’re already used as coefficients of the polynomial).

Since $ab>0$, $a$ and $b$ are either both positive or both negative; therefore $f'(x)=3ax^2+b$ is either always positive or always negative, i.e. it is never zero. If $f(x)$ had two distinct real roots, Rolle’s theorem would imply there was a point between the roots at which the derivative was zero – a contradiction. Hence the polynomial cannot have more than one real root.

Finally, note that the polynomial is cubic; therefore it must have at least one real root. Conclusion: it has exactly one real root.

Re: Using rolle's theorem to prove for roots

Olinguito said:
Your proof is essentially correct (although in one line you used $a$ and $b$ as roots of the polynomial whereas they’re already used as coefficients of the polynomial).

Since $ab>0$, $a$ and $b$ are either both positive or both negative; therefore $f'(x)=3ax^2+b$ is either always positive or always negative, i.e. it is never zero. If $f(x)$ had two distinct real roots, Rolle’s theorem would imply there was a point between the roots at which the derivative was zero – a contradiction. Hence the polynomial cannot have more than one real root.

Finally, note that the polynomial is cubic; therefore it must have at least one real root. Conclusion: it has exactly one real root.
Thanks, will change the usage of a and b accordingly.

## 1. What is Rolle's theorem and how is it used to prove for roots?

Rolle's theorem is a mathematical theorem that states that if a function is continuous on a closed interval and differentiable on the open interval, then there must be at least one point within the interval where the derivative of the function is equal to zero. This theorem is used to prove that a function has at least one real root within a given interval.

## 2. Why is Rolle's theorem important in mathematics?

Rolle's theorem is important because it provides a useful tool for proving the existence of roots for continuous and differentiable functions. It also serves as the basis for other important theorems in calculus, such as the Mean Value Theorem and the Fundamental Theorem of Calculus.

## 3. Can Rolle's theorem be used to prove the existence of multiple roots?

No, Rolle's theorem can only be used to prove the existence of at least one root within a given interval. It does not guarantee the existence of multiple roots.

## 4. How do you apply Rolle's theorem to a specific function?

To apply Rolle's theorem to a specific function, you must first check if the function satisfies the conditions of the theorem (continuous on a closed interval and differentiable on the open interval). Then, you can find the derivative of the function and set it equal to zero to solve for the point where the derivative is equal to zero. This point will be the root of the function within the given interval.

## 5. Are there any limitations to using Rolle's theorem to prove for roots?

Yes, there are limitations to using Rolle's theorem to prove for roots. It can only be applied to continuous and differentiable functions, and it can only prove the existence of at least one root within a given interval. It cannot be used to find the exact value of the root or to prove the existence of multiple roots.

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