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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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}=...
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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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Demonstration of arccos with Rolle and Lagrange
I am trying to solve this exercise from a secondary school textbook:
Prove that $\arccos x = \pi/2 - \arctan\bigl(x/\sqrt{1-x^2}\bigr)$ for $-1<x<1$.
I should use Lagrange or Rolle theorem for ...
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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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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 $...
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Trigonometric proof with calculus
How do we prove that $x^2=x\sin{x}+\cos{x}$ for exactly two values of $x$?
I thought of the sandwich theorem or such, getting only:
$-1\leq\sin{x}\leq1$
$-x\leq x\sin{x}\leq x$
$-x+\cos{x}\leq x\sin{x}...
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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 ...
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$f(x_0) + f''(x_0) = 2f'(x_0)$
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 ...
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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 $x$ ...
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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 $\xi \in (2, 4)$ ...
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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^{...
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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 $|a|\leq 2011$,$|b|\leq 2011$,$|...
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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 ...
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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 $\int_{0}^c (x-c)(f(x)-f(c)) dx = 0$.
...
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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 ...
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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 $f^{(n)}(x)\geq 0, \ \forall x\in \mathbb R$. ...
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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 ...
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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 ...
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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(...
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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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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'...
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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.
...
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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 $f'(c)=0$....
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$2019f'(x)+2020f(x)\geq2021$
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 ...
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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 &...
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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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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^{\...
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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)...
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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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$f:[a,b]\to\mathbb{R}$ 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{...
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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 ...
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If $3a+2b+3c=0,$ show that the cubic polynomial $ax^3+bx+c$ has at least one root in the interval $(0,2)$
In an old math forum (which was written about 15 years ago), I saw this question posed by a student and it had no answer. It intrigued me, so here it goes:
If $3a+2b+3c=0 \hspace{.1cm}(1),$ prove ...
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Showing there exists $\zeta$ such that $f(\zeta)=f'(\zeta)$ by Rolle's theorem: motivating the definition of an auxiliary function [closed]
Problem:
Let $f:[0,1] \rightarrow \Bbb R$ be a continous function that is differentiable on $(0,1)$, and where $f(0)=1$ and $f(1)=e$. Show that there exists $\zeta\in(0,1)$ such that $f(\zeta)=f'(\...
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Differential calculus using Rolles theorem and mean value theorem
Q- A function f is such that its second derivative is continuous on [a, a+h] and Derivable on (a,a+h). show that there exists a number $\theta$ between 0 and 1 such that-
$$ f(a+h) - f(a) -\frac{h[f'(...
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How do I determine further solutions of the equation using Rolle's theorem?
I gave this equation $2^x=1+x^2$ with the $1$st zero is $x_1=0$ and the $2$nd zero is $x_2=1$. (easy reading)
Now I want to calculate more zeros using Rolle's theorem, and I've rearranged the function ...
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Sum of roots for a even function and slope being 0?
High school senior and was helping my friend with Rolle's theorem and came across this neat point.
Given $f(x)$ is a continuous and differentiable function, and $f(-x)=f(x)$. Additionally, $f(x)=0$ ...
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Rolle's Theorem and error bound formula for trapezoidal rule
In these notes on error bounds
for numerical
integration,
it shows proofs for the error bound formula for the midpoint rule and the trapezoidal rule.
I don't understand the last part (page 5-6) for ...
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Derivative at end point of a interval
While studying Rolle's Theorem, a question came in my mind that can there exist a function which is continuous in the interval [a,b]
and differentiable in the interval (a,b) but not differentiable at ...
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Let $f(x)$ be non-constant, twice differentiable function define on $\mathbb R$ such that $y=f(x)$ is symmetric about line $x=1$
Let $f(x)$ be non-constant, twice differentiable function define on $\mathbb R$ such that $y=f(x)$ is symmetric about line $x=1$ and $f(-1)=f'(\frac{1}{4})=f'(\frac{1}{2})=0$ then which of the ...
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Is it possible to show the following without the assumption of continuity in a closed inteval?
Let $f(x)$ be diffrentiable at $(a,b)$ for $a,b\in \mathbb{R}$. given that $f'(x)\neq 0$ for any $x\in (a,b)$ proove that $f(x)=0$ for not more than one point in $(a,b)$
Well, If I was given that $f(...
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Problem involving Rolle's Theorem
Prove that, between any $2$ roots of $e^x\sin x = 1$, there exists a root of $e^x\cos x + 1=0$.
I am able to solve the problem in this way:
Let $a,b$ be the roots of $f(x)=e^x\sin x - 1$. Define $g(x)...
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