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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 ...
Cognoscenti's user avatar
2 votes
1 answer
83 views

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 ...
Kheerii's user avatar
  • 61
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0 answers
43 views

(LMVT) Minimum no. of roots of $f(x)(f'(x))^3+(f(x)^2)f'(x)f''(x)-f(x)f'(x)=x(f'(x))^2+xf(x)f''(x)-x$ based on some given conditions

Complete Question : Let $f(x)$ be a twice differentiable function such that $f(-1)=f(1)=1$ and $f(0)=0$. Find the minimum number of distinct solutions to the equation: $f(x)(f'(x))^3+(f(x)^2)f'(x)f''(...
Ayush Naman's user avatar
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0 answers
51 views

Attempt at an Analysis Question. Unsure of rigour

I am currently looking at differentiation in Real Analysis and attempting the following question: Suppose that $f,g : \mathbb{R} \rightarrow \mathbb{R}$ are functions with $f$ differentiable and $g(0)...
kodel's user avatar
  • 11
1 vote
1 answer
61 views

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 $f(c) = 2018\int_{0}^{c}f(x)dx$ ...
ten_to_tenth's user avatar
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1 vote
4 answers
110 views

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 ...
Maverick's user avatar
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1 vote
1 answer
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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. ...
Cinverse's user avatar
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1 answer
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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 ...
RobinSparrow's user avatar
  • 2,042
1 vote
0 answers
43 views

Is there a missing assumption in this problem?

I believe this problem is missing an assumption. We have two differentiable functions, $f$ and $g$, which both vanish at $0$. Let $t > 0$ with $f(t), g(t) > 0$. Define $h(x) = f(t) g(x) - g(t) f(...
Valor Vaporeon's user avatar
2 votes
2 answers
153 views

Let $f$ be continuous in $[0,2]$ and differentiable in $(0,2)$ such that $|f'(x)|\leqslant1$ and $f(0)=1=f(2)$. Prove that $f(x)\geqslant0$.

Let $f$ be a continuous function in $[0,2]$ and differentiable in $(0,2)$, such that $|f'(x)|\leqslant1$ for all $x\in(0,2)$ and also $f(0)=1=f(2)$. Prove that $f(x)\geqslant0$ for all $x\in(0,2)$. My ...
MATH14's user avatar
  • 347
1 vote
1 answer
161 views

Rolle's theorem: if $f(0)=f(\pi)=0$ then $f'(c)=\tan(f(c))$ for some $c$.

Let $f$ be a continuous function going from $[0, \pi]$ to $(-\pi/2,\pi/2)$. I know that $f$ is differentiable and $f(0)=f(\pi)=0$. Prove that there exists $c$ in $(0,\pi)$ so that $f'(c)=\tan(f(c))$. ...
Ngân Kim's user avatar
6 votes
3 answers
364 views

Rolle's theorem? [duplicate]

The function $f$ is differentiable in $[0,1]$ and $f$ has infinite roots in $[0,1]$. Prove that there exists $c\in [0,1]$ such that $f(c)=f'(c)=0.$ My attempt: Assume $x_1, x_2,...,x_n,...$ are roots ...
user avatar
1 vote
2 answers
74 views

Finding another root to apply Rolle's theorem.

Let $f(x)$ be continuous on $[0,+\infty]$, continuously differentiable on $(0,+\infty)$ and $f(0)=1$; $\underset{x\to +\infty}{\lim} f(x)=0.$ Prove that there exists $c> 0$ such that $f'(c)+e^{-c}=...
user avatar
2 votes
0 answers
70 views

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 ...
CESTU's user avatar
  • 33
1 vote
1 answer
87 views

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 ...
Michele Assirelli's user avatar
2 votes
3 answers
143 views

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 ...
gem's user avatar
  • 21
3 votes
0 answers
80 views

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 $f'(\alpha)=0$. ...
Archisman 's user avatar
1 vote
2 answers
246 views

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 ...
Thomas Finley's user avatar
0 votes
1 answer
67 views

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 ...
user19872448's user avatar
4 votes
1 answer
282 views

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 ...
IkerUCM's user avatar
  • 402
0 votes
1 answer
81 views

Solve problems of the form "given a differentiable function prove that there exists a point c such that"

The problems in question all have essentially the same setup: Given a function $f$ that is continuous on $[a,b]$ and differentiable on $(a,b)$ prove that there exists a point $c \in (a,b)$ such that $...
evolved_antenna's user avatar
3 votes
1 answer
319 views

Application of Rolle's theorem in real analysis

If $f$ is continuous on $[-2,2]$ and thrice differentiable on $(-2,2)$ and $f(2)=-f(-2)=4$ and $f'(0)=0$ then there exist $x\in(-2,2)$ such that $f'''(x)\ge3$ I have solved that We know that $f$ is ...
Abcd's user avatar
  • 447
3 votes
1 answer
60 views

$f(x_0) + f''(x_0) = 2f'(x_0)$

f $\in$ C$^2$[a,b], and f has at least three distinct roots in [a,b]. I'm required to show that there's a point x$_0$ in [a,b], such that f(x$_0$) + f''(x$_0$) = 2f'(x$_0$). I concluded that there are ...
Arfin's user avatar
  • 1,445
0 votes
1 answer
57 views

How to adapt this proof of l'Hospital's rule to the case $\lim f(x)= \infty = \lim g(x) $

I am looking for an adapted version to the case where $\lim_{a^+} f = \infty$ and $\lim_{a^+} g = \infty$ of this proof. It goes like this: Suppose $f'(x), g'(x)$ exist and $g'(x) \neq 0$ for all $x$ ...
niobium's user avatar
  • 1,231
1 vote
1 answer
55 views

Existence of $\xi$

Let $f(x) = \left\{\begin{matrix} e^{\frac{\ln x}{x}} &, & x>0 \\ 0 & , & x = 0 \end{matrix}\right.$. If $F$ is a primitive of $f$ then prove that there exists $\xi \in (2, 4)$ ...
Tolaso's user avatar
  • 6,686
0 votes
1 answer
54 views

Prove that $ f([0,1]) = [0, \frac{1}{e}] $

Let $ f(x) = xe^{-x} $. I need to prove that $ f([0, 1]) = [0, \frac{1}{e}] $. Can you verify my proof? $ \underline{[0, \frac{1}{e}] \subseteq f([0, 1]):} $ First, $ f(0) = 0e^0 = 0, f(1) = 1\cdot e^{...
talopl's user avatar
  • 1,009
1 vote
1 answer
84 views

Prove location of roots

So I was tasked with the following problem: Given that the following equation is quadratic and has a real root: $$ax^2+bx+c=0$$ Prove that if a,b,c $\in \mathbb{Z}$ and $|a|\leq 2011$,$|b|\leq 2011$,$|...
Helixglich's user avatar
0 votes
1 answer
91 views

Prove that there exists $d$ in$(a,b)$ such that $f'''(d)=0$

Let $f$ be a non-negative function which is three-times differentiable on $(a,b)$. If there exist two numbers $c_1$,$c_2$ in $(a,b)$ with $c_1<c_2$ such that $$f(c_1)=f(c_2)=0,$$ proved that there ...
tms's user avatar
  • 3
10 votes
2 answers
699 views

Prove the existence of a point $c$.

Problem Let $f$ be a continuous function, $f:[0,1]\to\mathbb{R}$ with $\int_{0}^1 (2x-1)f(x) dx = 0$. Prove that there exists a point c between $(0, 1)$ such that $\int_{0}^c (x-c)(f(x)-f(c)) dx = 0$. ...
Mogovan Jonathan's user avatar
0 votes
1 answer
402 views

Why do we need continuity at the end points of the interval for Rolle's theorem?

For a function ($f:[a,b] \to \Bbb R$) to satisfy Rolle's theorem it must be continuous in the interval $[a,b]$, differentiable in $(a,b)$ and $f(a)=f(b)$. I don't really understand what's the need of ...
Ankit's user avatar
  • 710
2 votes
1 answer
80 views

A function with weakly positive $n$-th derivative has at most $n$ roots – an inequality version of the Fundamental theorem of Algebra.

This is an attempt to generalize the result in [1]. Claim: Let $n\in \mathbb N$, and let $f:\mathbb R \to \mathbb R$ be such that its $n$-th derivative $f^{(n)}(x)\geq 0, \ \forall x\in \mathbb R$. ...
Pavel Kocourek's user avatar
8 votes
1 answer
304 views

A function with positive $n$-th derivative has at most $n$ roots – an inequality version of the Fundamental theorem of Algebra.

Claim: Let $n\in \mathbb N$, and let $f:\mathbb R \to \mathbb R$ be such that its $n$-th derivative $f^{(n)}(x)>0, \ \forall x\in \mathbb R$, then $f$ has at most $n$ roots. Context: The ...
Pavel Kocourek's user avatar
1 vote
0 answers
103 views

Rolle' Theorem on Second Derivative

Suppose that the function $f$ has second derivative in the interval $(a,b)$ and that $f(x_1) = f(x_2)=f(x_3)$, with $a < x_1<x_2<x_3<b$. Then, there exist a $\epsilon \in (a,b)$, such that ...
Niccolo's user avatar
  • 694
5 votes
3 answers
337 views

Show that a function has exactly one real root

I have a function : $$f(x) = e^{3x}+4e^{2x}-3e^x=0$$ I have to show that there is exactly one real root. My effort $f$ is continuous on $\mathbb{R}$ $f(-1) = e^{-3}+4e^{-2}-3e^{-1} \approx =-0.512$ $f(...
Homer Jay Simpson's user avatar
2 votes
1 answer
216 views

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 ...
Niccolo's user avatar
  • 694
1 vote
1 answer
98 views

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 ...
Delta Dsr's user avatar
3 votes
2 answers
248 views

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 ...
Lee Wei Xuan's user avatar
3 votes
0 answers
64 views

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 ...
Maverick's user avatar
  • 9,599
1 vote
1 answer
69 views

Intuitive understanding for $f(x)=f'(x)$ in $(a,b)$ when $f(a)=f(b)=0$

Question: Let $f(x)$ be a differentiable function and $f(a)=f(b)=0 \;(a\lt b)$ then in the interval $(a,b)$ $f(x)+f'(x)=0$ has at least one root. $f(x)-f'(x)=0$ has at least one root. $f(x)\times f'...
aarbee's user avatar
  • 8,338
0 votes
0 answers
20 views

How to do this question with the help of Rolle's theorm [duplicate]

If $a_1$, $a_2$, $a_3$,$\cdots$ $a_n$ $(n\ge2)$ are real and $(n-1){a_1}^2-2na_2<0$ then prove that at least two roots of the equation $$f(x)=x^n+a_1x^{n-1}+a_2x^{n-2}+\cdots+a_n=0$$ are imaginary. ...
user avatar
0 votes
1 answer
88 views

a little problem about Rolle theorem

As Rolle theorem goes,if $f(x)$ is continuous and well-defined in $[a,b]$, derivable in $(a,b)$, and $f'(x)$ is bounded, $f(a)=f(b)$, then there exists $c$ ($a<c<b$), which satisfies $f'(c)=0$....
xinyi's user avatar
  • 13
3 votes
1 answer
95 views

$2019f'(x)+2020f(x)\geq2021$

Find all continuous function $f:[0,1]\rightarrow\mathbb{R}$ which is differentiable on $(0,1)$ and $$f(0)=f(1)=\frac{2021}{2020}\textrm{ while }2019f'(x)+2020f(x)\geq2021,\forall x\in(0,1).$$ The ...
MyName'sJeff's user avatar
1 vote
0 answers
36 views

Application of Rolle's theorem (a corollary of the mean value theorem of differentiation) [duplicate]

The following task is given: Let $n \in \mathbb{N}$ and $f$ be a $n-$times differentiable function. With $f^{(n)}$ we name the nth derivation of $f$. Show: If there exist $n+1$ different numbers $x_1 &...
X3nius's user avatar
  • 55
2 votes
0 answers
55 views

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 ...
Orion_Pax's user avatar
  • 431
0 votes
0 answers
73 views

Proof checking for Rolles Theorem

This was a proof given for Rolles Theorem : Let $f$ be continuous on $[a, b], a<b$, and differentiable on $(a, b)$. Suppose $f(a)=$ $f(b)$. Then there exists $c$ such that $c \in(a, b)$ and $f^{\...
Orion_Pax's user avatar
  • 431
0 votes
1 answer
562 views

Let $a$ be an integer such that all real roots of the polynomial $2x^5+5x^4+10x^3+10x^2+10x+10$ lie in the interval $(a,a+1)$

We need to find what $a$ is. Can I not solve this using the mean value theorem in the interval $(a,a+1)$? I tried for a $c\in(a,a+1)$ $$f(c)=\int_a^{a+1}f(x)dx$$ but since all the roots lie in $(a,a+1)...
Parth Shresth's user avatar
3 votes
2 answers
72 views

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 $\underset{x\rightarrow-\infty}{\lim}f(x)=L$ ...
G. Ticher's user avatar
  • 135
3 votes
2 answers
112 views

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 ...
Ninja's user avatar
  • 2,807
0 votes
2 answers
812 views

$f:[a,b]\to\mathbb{R}$ continuous function and differentiable in (a,b) $\rightarrow$ $\exists$ $c\in(a,b)$ with $f(c)=\frac{1}{a-c}+\frac{1}{b-c}$

I am preparing for my exam and need help for the following task: Let $f:[a,b]\to\mathbb{R}$ be a continuous function and differentiable in (a,b). a) Show that $c \in(a,b)$ exist such that $f(c)=\frac{...
Analysis_Mark's user avatar
0 votes
1 answer
176 views

Number of roots of $2013x^{2012}-2012x^{2011}-16x+8=0$ in $[0,8^\frac1{2011}]$

If $P(x)=2013x^{2012}-2012x^{2011}-16x+8$, then $P(x)=0$ for $x\in\left[0,8^{\frac{1}{2011}}\right]$ has exactly one real root. no real root. at least one and at most two real roots. at least two ...
aarbee's user avatar
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