# Questions tagged [rolles-theorem]

For questions about Rolle's Theorem, or exercises that suggest the use of Rolle's Theorem. The theorem states that if a real-valued differentiable function has two distinct zeros, then the function has a vanishing derivative for some value between those two zeros.

99 questions
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### Proving $\sin(x) < x$ for $0<x<2\pi$

Prove that $\sin(x) < x$ when $0<x<2\pi.$ I have been struggling on this problem for quite some time and I do not understand some parts of the problem. I am supposed to use rolles theorem ...
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### Proving a lemma for a dichotomy

I am currently trying to prove the Rolle theorem by dichotomy. I think it is possible and I found the following property, that might be useful : let $f : [a,b]\to \mathbf{R}$ be a continuous ...
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### Limits of Rolle theorem

I would like to see a function $f:[a,b]\to\mathbb{R}$ that is differentiable in $(a,b)$ but it is not continuous at least at one of the interval boundary points $a$ or $b$. Can you show me one? This ...
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### Let $f:\mathbb R\to \mathbb R$ be differentiable such that $f'(x)>f(x)$ for all $x\in \mathbb R$ and $f(0)=1$, then $f(1)$ lies in which interval?

Let $f:\mathbb R\to \mathbb R$ be differentiable function such that $f'(x)>f(x)$ for all $x\in \mathbb R$ and $f(0)=1$, then $f(1)$ lies in which one of the intervals ? a)$(0,e^{-1})$ ...
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### The intermediate value property

I want to prove this statement, Assume $f:(a,b)\to \mathbb R$ has intermediate value property, then $f$ cannot have jump discontinuities. So, i have two way to prove; assume $f:(a,b)\to \mathbb R$ ...
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### Can we conclude that $f$ is continuous on the interval $[a,b]$?

Corollary $1$. If $f$ is differentiable on the interval $a<x<b$, then the zeros of $f$ are separated by zeros of $f'$. proof. Let $x_1,x_2,...,x_n$ be the real roots of the function $f$. ...
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### Rolle's Theorem Function [closed]

Find all numbers, $c$ that satisfies the conclusion of Rolle's Theorem for the following function, $f(x)=x^2−10x+10,[0,10]$ I haven't learned this theorem yet and am confused on what to do.
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### Rolles theorem used for solving equation $ax^3+bx^2+cx+d=0$

If a,b,c,d are Real number such that $\frac{3a+2b}{c+d}+\frac{3}{2}=0$. Then the equation $ax^3+bx^2+cx+d=0$ has (1) at least one root in [-2,0] (2) at least one root in [0,2] (3) at least two ...
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### If Rolle's Theorem is assumed to be true, doesn't that prove the MVT?

If we assume that Rolle's Theorem is true is it practical to say that it also proves the MVT? My reasoning is that even though Rolle's Theorem is the special case for when $f(a)=f(b)$ and the secant ...
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### How to use mean value theorem to derive this Taylor expansion equation?

The mean value theorem says there exists a $c$ in $(a, b)$, such that $f(b)-f(a)=f'(c)(b-a)$. There should also be equation for second order, $f(b)-f(a)=f'(a)(b-a)+1/2*f''(d)(b-a)$ where $d$ is ...
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### Rolle's theorem on $e^x\sin x-1=0$

Prove that between any two real roots of the equation $e^x\sin x-1=0$ the equation $e^x\cos x+1=0$ has at least one root. My attempts: By Rolle's theorem, the derivative of $e^x\sin x-1=0$ has at ...
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### Showing point in f(x) such that f'(x)=0 exists [closed]

let $f(x)=x^4 + \sin x$ Show that there exists $x \in (-2,2)$ such that $f'(x) = 0$.
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### How to use Rolle's Theorem to show that $f$ has at most one fixed point? NOT DUPLICATE [duplicate]

Let $a,b \in \mathbb{R}$ be such that $a\lt b$. Suppose that $f:[a,b]→\mathbb{R}$ is a continuous function on $[a,b]$ and is differentiable on $(a,b)$ and that $f'(x) \gt 1$ , for all $x\in(a,b)$. ...
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### Presenting a lecture on Mean value theorems.

I've to give a model lecture on Mean Value theorems(Rolle's theorem,Lagrange's Mean value theorem,Cauchy's Mean value theorem).But i do not know what content i should include in the lesson(I do not ...
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### Rolle's theorem: what's the right statement of the theorem?

In the fourth edition of "Introduction to Real Analysis" by Bartle and Sherbert, theorem 6.2.3 (Rolle's theorem) states, Suppose that f is continuous on a closed interval $I := [a, b]$, that the ...
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### Application of Derivatives (Rolle's Theorem?)

Let there be a cubic equation $f(x)=ax³+bx²+cx+d=0$, and the coefficients of the equation be related by $-a+b-c+d=3$ and $8a+4b+2c+d=6$. How can I show that the quadratic equation $3ax²+2bx+c-1$ has a ...
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### Rolle's theorem without compact domain hypothesis [duplicate]

Good evening everyone, I'm asking for a proof of "Rolle's theorem generalization". The thesis is as follows: Let be $a \in \mathbb{R}$ and $f:[a,\infty) \longrightarrow \mathbb{R}$ a continuous ...
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### Proving Rolle's Theorem

Trying to prove Rolle's Theorem, which says that for a function $f$ continuous over $[a,b]$ and differentiable over $(a,b)$ (no idea why the endpoints aren't included here), such that $f(a) = f(b)$, ...