# 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.

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### Problem 237 "Mathematical Quickies:270 Stimulating Problems with Solutions" Particle Movement

I was solving "Mathematical Quickies:270 Stimulating Problems with Solutions" when I came across a very peculiar question (Problem 237): A particle moves in a straight line starting from ...
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### Let $f(x):\mathbb{R}\to [-1,1]$ be twice differentiable and $f(0)^2+f'(0)^2=4$, then p.t. $\exists x_0$ s.t. $f(x_0)+f''(x_0)=0$ but $f'(x_0)\ne 0$ [duplicate]

The actual question is a multiple correct MCQ, but this was the only part I was having trouble with. I also can't fully attest for the correctness of the question, although my answer key does put this ...
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### Differentiable $f: [0, 1] → \mathbb{R}: \int_{0}^{1} f(x)dx = \int_{0}^{1}xf(x)dx.$ Prove $\exists c \in (0, 1):f(c) = 2018\int_{0}^{c}f(x)dx$

$f: [0, 1] \rightarrow \mathbb{R}$ is a differentiable function such that $\int_{0}^{1} f(x)dx = \int_{0}^{1}xf(x)dx.$ Prove that there exists $c \in (0, 1)$ such that $f(c) = 2018\int_{0}^{c}f(x)dx$ ...
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### If $f(a)=b$ and $f(b)=a$. Prove that there exists at least one $c$ such that $|f'(c)|<1$. Also prove that ther exists some 'd' such that $|f'(d)|>1$

Let $f:[a,b]\rightarrow[a,b]$ where $a<b$ be a non-linear differentiable function such that $f(a)=b$ and $f(b)=a$. Prove that there exists at least one $c$ such that $|f'(c)|<1$ .Also prove that ...
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### (Re)Defining Rolle's theorem and its converse

My textbook has stated Rolle's theorem as: Let $f : [a,b] \to \mathbb{R}$ be continuous on $[a,b]$ and differentiable on $(a, b)$, such that $f(a) = f(b)$, where $a$ and $b$ are some real numbers. ...
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### Assumptions underlying Rolle's Theorem

Edit: this question reflects some sloppy thinking about derivatives, now corrected. I left the question untouched in case others may benefit. The point of clarification is that if a derivative exists ...
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### How to show that there exists $\xi \in (a,b)$ such that $f'(\xi)=\frac{f(\xi)-f(a)}{\xi-a}$ [duplicate]

Given that $f(x)$ is differentiable on the closed interval $[a,b]$ and $f'(a)=f'(b)$.How can I prove that there exists $\xi \in (a,b)$ such that$$f'(\xi)=\frac{f(\xi)-f(a)}{\xi-a}$$ I have tried to ...
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### Using Rolle's Theorem to prove roots

I have to prove using Rolle theorem that the equation $x^3-3x+4=0$ does not have more than one solution in $[-1,1]$. By looking at similar problems (here for example) i supposed that the equation does ...
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### Rolle's Theorem in proving exactly one real zero

The question asks to use the Rolle's Theorem to prove $f(x)=(x-8)^3$ has only one real zero. I have already used IVT to prove that there is a zero, but I'm stumped on how to use the Rolle's Theorem to ...
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### How to proof the derivative is continuous

Q) Consider a function $f:(0,1) \to \mathbb{R}$, continuous annd differentiable. $(\alpha_n)^\infty_1$ be a sequence of roots of $f$ which converges to $\alpha \in (0,1)$. Prove that $f'(\alpha)=0$. ...
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### Is this proof of Rolle's Theorem a valid one?

Let $f:\Bbb [a,b]\to \Bbb R$ satisfy the following : (i) $f(x)$ is continuous in $[a,b]$ (ii) $f(x)$ is derivable in $(a,b)$ (iii) $f(a)=f(b)$ then, $\exists c\in (a,b)$ such that $f'(c)= 0.$ I tried ...
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### How to choose functions in proofs type Rolle?

while I was solving some exercises related to Rolle's theorem or the intermediate value theorem I realized that the trick is always knowing how to choose the function, for example, to show that every ...
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### Prove $x^2 + x \cos (x) -2 \cos ^2 (x) = 0$ has exactly two real roots

Given the equation $$x^2 + x \cos (x) - 2 \cos ^2 (x) = 0$$ prove it has exactly two real roots. My attempt In order two prove the equation indeed just has two real roots, first of all I need to ...
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### On Rolle's Theorem

Assuming that the function $f$ is differentiable in $(0,1)$ and continuous on $[0,1]$. If $f(1) = 0$, show that there exists one $c \in (0,1)$, such that $$f(c) = \frac{c f'(c)}{100}.$$ My Attempt: I ...
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### discovering Mean Value Theorem

mean value theorem for single variable function is very easy and intuitive once you "see" the formula. Actually, My question, slightly weird but helpful, is that How does someone come up ...
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### Prove that there exists $\xi \in (a,b)$ such that $f(a)-2f(\frac{a+b}{2})+f(b)=\frac{1}{4}(b-a)^2f''(\xi) .$ [duplicate]

Given that f is twice differentiable on $[a,b]$, prove that there exists $\xi \in (a,b)$ such that $$f(a)-2f\left(\frac{a+b}{2}\right)+f(b)=\frac{1}{4}(b-a)^2f''(\xi) .$$ This problem was given in a ...
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### If for $x\in(\frac{1}{2},\infty)$ we have $f'(x)=(e^x-1)(x-2)(x-3)$. Show that there exist exactly two roots of $f''(x)=0$ in the given domain

Let $f:\left(\frac{1}{2},\infty \right)\to \mathbb{R}$ be a function such that $f'(x)=(e^x-1)(x-2)(x-3)$. Show that there exist exactly two roots of $f''(x)=0$ in the given domain. My Attempt I ...
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### Rolles theorem: differentiabilty open brackets significance

https://math.stackexchange.com/a/2863979/922054 Related to Rolle's theorem : Here it is said that $f(x)$ is non-differentiable at endpoints of $\sqrt{1-x^2}$, but isn't it okay to talk of endpoint ...
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### If $f(x)\rightarrow L$ from both sides then $f'(c)=0$ for some $c$ [duplicate]

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f$ is differentiable over $\mathbb{R}$. Prove that if $\underset{x\rightarrow\infty}{\lim}f(x)=L$ and $\underset{x\rightarrow-\infty}{\lim}f(x)=L$ ...
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### Finding the number of real roots of a polynomial

I want to find the number of real roots of $f(t)=t^4-2t^2+4t+1$. As $f(0)=1>0, f(-1)=-4 <0$ and $f(-2)=1>0$, I can say that there are two real roots since the polynomial is continuous. For ...
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