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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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 show a specific complicated function has at most one root

Let $f(x) = \lambda_1^{x+1} \cdot \frac{\log(\lambda_1)}{1-\lambda_1} \cdot \left((\lambda_1 + \lambda_2 - 1) \cdot \log(1 + \frac{1}{\lambda_1 + \lambda_2 - 1})\right) - \sum\limits_{i=1}^\infty \...
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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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Integral and number of solutions

I would need some help with these 2 exercises: sin(x)*sin(2x)=1 and x is from 0 to 4pi. Number of solutions for this ecuation. 2 integral from 0 to pi/2 from 1/(sin(x + pi/6)*sin(x+pi/3))
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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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Uniqueness of the ODE $-c* u''(x) + (u(x)')^2 = 1$

I'm struggling to show that there's a unique solution $u\in C^2(-1,1)\cap C[-1,1]$ for the Dirichlet / Boundary Value problem $-c* u''(x) + (u(x)')^2 = 1$ in $(-1,1)$ and $u(-1) = 0 = u(1)$, where $c&...
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$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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4 votes
1 answer
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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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4 votes
1 answer
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Existence of $x_0 \in (0,1)$, s.t. $f'(x_0)+1/x_0 f(x_0)=2$

$f(x)$ is differentiable on $[0,1]$, $\int_{0}^{1}{f(x)}dx=1/2$, how to imply exist $x_0 \in (0,1)$, s.t. $f'(x_0)+1/x_0 f(x_0)=2$ I think it is equal to prove $\exists x_0, s.t. (x_0f(x_0))'=2x_0$ I ...
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Prove there exists a point $\xi$ so that $f'(\xi)=0$

If we have a function $f(x)$ is differentiable in $\mathbb{R}$ and $\lim_{x\rightarrow \pm \infty} f(x) = 17 $. How can we show that $\exists \xi \in (-\infty, \infty) | f'(\xi) = 0$? I'm wondering ...
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1 vote
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Mean value theorem to prove existence of zero for second derivative [duplicate]

I'm going to try to solve the problem presented below. But as you will realize, I'll run into a problem. I hope you can give me somet tips. Let $f(x)$ be a function which is contiuous on $[a, b]$ and ...
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2 votes
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Let $f$ be a continuous function in $[0,5]$ and twice differentiable function in $(0,5)$ such that $f(4)=f(5)=0$. Prove the following

Let $f$ be a continuous function in $[0,5]$ and twice differentiable function in $(0,5)$ such that $f(4)=f(5)=0$. Prove the following: There exists some $a$ in $[0,5]$ such that $nf(a)+af'(a)=0, n\in ...
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If $P_n(x)$ has n real roots, then it's derivative has no complex roots

Usually, I would use Rolle's Theorem. Every polynomial is continuous and differentiable. Using Rolle's Theorem, derivatives of functions have $n-1$ real roots: Between every two consecutive roots $a$ ...
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2 votes
1 answer
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Prove that the polynomial has exactly 2 real roots by IVT or Rolle's Theorem

I'm trying to prove this by IVT or Rolle's Theorem. Usually if it would say "Prove it has only 1 real root", I would assume it had 2 roots, take the derivative and if it was $≠0$, then I ...
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2 votes
1 answer
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Solving the equation of the type $g\left( x \right) = \int\limits_0^x {f\left( t \right)dt} $

Let $f:\left[ {0,1} \right] \to R$ be a differentiable function. Let $g\left( x \right) = \int\limits_0^x {f\left( t \right)dt} $ with $g\left( 1 \right) = 0$ . Which of these equations must have at ...
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Interval in which roots of given function lies is:

Let $f(x)=f\left(x\right)=\frac{x}{\sin x}$ & $x∈\left(0,\frac{\pi}{2}\right)$, Then prove that the interval in which at least one root of equation $h\left(x\right)$= $\frac{2}{x-f\left(\frac{\pi}{...
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2 answers
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If $f$ is twice differentiable such that $f\left(\frac{1}{n}\right)=0$, then what can we say about $f''(0)$?

The continuity of $f, f'$ and Rolles theorem implies that $f(0)=0$ and $f'(0)=0$, but is it true that $f''(0)$. We have a sequence $(x_n)$ such that $f''(x_n)=0$ and $x_n \to 0$. But continuity of $f''...
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$f(x) = (x^2 -1)^2(a_0 x^3+a_1 x^2 + a_2 x + a_3 )$ $f'(x)$ has exactly $3$ distinct real roots and $f''(x)$ has two distinct real roots. Find $f(x)$.

Can we get a function in the following form and properties ? $f(x) = (x^2 -1)^2(a_0 x^3+a_1 x^2 + a_2 x + a_3 )$ $f'(x)$ has exactly $3$ distinct real roots and $f''(x)$ has two distinct real roots. ...
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Verify Rolle's theorem

Verify Rolle's theorem for the function $f(x)=x^2+5x-6$ in the interval (-6,1) After checking continuous for the function. I saw the function is continuous and differentiable. After checking ...
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1 answer
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Rules of checking differentiability for Rolle's theorem

I was doing some exercises of Rolle's theorem. But, they didn't check the differentiability the way we checked differentiability normally. I am giving some examples. When I was checking ...
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1 vote
1 answer
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Is Rolle's theorem and Mean values theorem same?

You are correct in saying that these theorems are essentially the same. The mean value theorem is a general form of the Roll's theorem where the slope of secant is not necessarily zero. Both theorems ...
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What's the meaning of different braces?

Since the function $f(x)$ is continuous in the closed interval $[a,b]$ and differentiable in the open interval $(a,b)$, so the function $F(x)$ will be continuous in the interval $[a,b]$ and satisfied ...
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Range of values of 'a' so that the function has local maxima/minima at given values of x

Find the set of all the possible values of $a$ for which the function: $$f(x)=5+(a-2)x+(a-1)x^2-x^3$$ Has a local minimum value at some x<1 and local maximum value at some x>1. I started by ...
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1 vote
0 answers
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Let $f(x)$ be a function such that $f(x)=f(4-x)$ , then find the minimum number of roots of $f''(x)=0$ in $[0,4]$.

Let $f(x)$ be a non-constant twice differentiable function defined on $(-\infty,\infty)$ such that $f(x)=f(4-x)$ and $f(x)=0$ has at least two distinct repeated roots in $(2,4)$, then find the minimum ...
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1 vote
1 answer
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If $f(a) = f(b)$ and $f'(a) = f'(b)$, prove that for every real number $λ$ the equation $f''(x) − λ(f'(x))^2 = 0$ has at least one solution in $(a,b)$

Let $f : [a, b] → R$ be a function, continuous on $[a, b]$ and twice differentiable on $(a, b)$. If $f(a) = f(b)$ and $f'(a) = f'(b)$, prove that for every real number $λ$ the equation $f''(x) − λ(f'(...
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Upper limit of differential connection to original function

I have a homework with a question that goes like this: Let $f(x)$ be a differentiable function at $[a,b]$. assuming: $f(a)=f(b)=0$ $f(x)>0$ for all $x$ in $(a,b)$ there is a M that: $|f^′(x)|\le M$...
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1 vote
1 answer
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Intuitive approach towards the auxiliary function of Lagrange's mean value theorem proof

I have often seen many people using another auxiliary function $g(x)$ such that $g(x)=f(x)-\frac{f(b)-f(a)}{b-a}\cdot x$ where $f(x)$ is our original function continuous in $[a,b]$, differentiable in $...
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-1 votes
1 answer
61 views

Change in the number of positive zeros of a continuous function.

Let $f(x)$ be any continuous function, then is it true that $$Z^{+}\left(\alpha f(x)+(x+\beta)f'(x)\right)\leq Z^{+}\left(f'(x)\right)+1$$ where $\alpha>1$ is a real number and $\beta$ is any ...
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0 votes
2 answers
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Show that $p(D) f(x)$ has at least one zero on $[a, b]$, where D denotes $\frac{d}{dx}$.

Let $p(x)$ be a real polynomial of degree n with leading coefficient 1 and all roots real. Let R be the reals and $f : [a, b] → R$ be an n times differentiable function with at least $n + 1$ distinct ...
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-1 votes
1 answer
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Is there a conflict between an Euler's ODE and Rolle's theorem here? [closed]

Let a polynomial $f(x)$ of degree $n>2$ or more has $n$ number of distinct non-zero roots then we can prove that the equation $$x^2f''(x)+3xf'(x)+f(x)=0~~~~(1)$$ has at least $n$ real root. Let $g(...
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1 vote
3 answers
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Comparing $e^{4}-2$ and $50$ without calculator

I'm required to compare $e^{4}-2$ and $50$ without using calculator. I thought of the following way: Let a function $h(x)$ be defined as $e^{x}-13x.$ If I can prove that this function at $x=4$ is ...
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0 votes
1 answer
125 views

Absolute of Second Derivative of f is less than or equal to M

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a twice differentiable function Suppose $|f''(x)| \leq M$ on $[a, b]$. Show that for all $x \in [a, b]$ $$|f'(x)| \leq \left|\frac{f(b)-f(a)}{b-a}\right|+(b-...
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0 votes
0 answers
51 views

Roll mean value theorem prove equation

Assume that $f(x)$ is continuous in $[a,b]$ and differential in $(a,b)$, and $f(a)=f(b)=0$, prove that there exists a $\xi\in(a,b)$, s.t $f'(\xi)=f(\xi)$.
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