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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What is the correct formulation of Rolle's Theorem

Generally Rolle's theorem is expressed with words. But how should it look in a formal mathematical/logic language? $$\forall f \Big( \big(f \textrm{continue [a,b]} \land f \textrm{differentiable ]a,b[...
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Rolle's theorem and inequality

Let $f$ be continuous on $[0,1]$ and differentiable on $(0,1)$ Suppose $f(0)=f(1)=0$ and $\exists\texttt{ } x_{0}$ such that $f(x_{0})=1$ Prove that $$\left|f'(c)\right| \gt 2$$ for some $c \in(0,1)$ ...
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32 views

Rolle's theorem for a equation

Let $f:R \to(0,+\infty)$ be a differtiable function, and $f(1)=1$. Prove that there is at least on $ξ \in (0,1)$ s.t. $$f(ξ)=\frac{1}{e^\frac{ξf'(ξ)}{f(ξ)}}$$ My try was to maybe use Rolle's theorem. ...
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Rolles Theorem, IVT, Mean Value theorem [closed]

How to prove function $g ~\colon (0, 2) \to \mathbb{R}$, given by $g(x)=x^3+x-3$ cannot have two zeros. Deducing that the function has exactly one zero. Edit: Note that domain is in $\mathbb{R}$, ...
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Why do I get wrong result from Rolle's theorem in the given functions?

While solving this question I ran into some trouble. Consider using Mean value theorem we have : $f(1)= 6$ and $f(0)=2$ Thus, $$ \frac {f(1)-f(0)}{1-0}=f'(c_1) $$ for some $ 0 \lt c_1 \lt 1$ And ...
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19 views

any concrete relationship between roots of polynomial and coeefficients+special pattern [duplicate]

if $a_0$, $a_1$, ..,$a_n$ are real numbers such that $a_0+\frac{a_1}{2}+\frac{a_2}{3}+\frac{a_n}{n+1}=0$ prove the polynomial $a_0+a_1x+...+a_nx^n=0$ has a solution between 0,1. This is a problem ...
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1answer
52 views

$f: [0,1] \rightarrow \Bbb R$ be a differentiable non-constant funtion such that $f(0) = f(1)$.

$f: [0,1] \rightarrow \Bbb R$ be a differentiable non-constant funtion such that $f(0) = f(1)$. Show $\exists$ a point $x \in [0, 1]$ such that $f'(x)$ is rational. Here's what I've done: I ...
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Question on Rolle's Theorem

Let $f: [a, b] \rightarrow \mathbb{R}$ be a function, continuous on$[a, b]$ and twice differentiable on $(a, b)$. If $f(a) = f(b)$ and $f'(a) = f'(b)$. Then find the least number of roots of the ...
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48 views

Does Rolle's theorem apply for roots at infinity? [duplicate]

Consider the function $$f(x) = \frac 1 {x^2+1}$$ $f(x) \to 0$ as $x \to \pm\infty$, thus we can say that $f(x)$ has roots at $\pm\infty$. Can we apply Rolle's theorem to prove $f'(c)=0, c \in (-\infty,...
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Prove that there exists a line passing through M$(\alpha,\beta)$ which is tangent to the graph of $f$.

Let $f:[a,b]\rightarrow\Bbb{R}$ be a function continuous on $[a,b]$ and differentiable on $(a,b)$. Let M$(\alpha,\beta)$ be a point on the line passing through the points $(a,f(a))$ and $(b,f(b))$ ...
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37 views

Use Rolle to determine the number of possible solution of $f(x) = \sin x + x + 1$

Good morning. I've found an exercise that require to find the number of possible solutions of the equation $f(x) = \sin x +x + 1$ in $[0, 2\pi]$ using Rolle's theorem. How should I proceed? I know ...
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Prove that $\nexists c \in ]-1,1[$ such that $f'(c) = 0$

Given the function $f(x) = 1 - |x|^{2/3}$ which satifies that $f(-1) = f(1) = 0$ show that$\nexists c \in ]-1,1[$ s.t. $f'(c) = 0$. Furthermore I have to conclude why this does not contradict Rolle's ...
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Prove that there exists some $\xi\in(-\frac{\pi}{2},\frac{\pi}{2})$ such that $f''(\xi)=f(\xi)(1+2\tan^{2}{\xi}).$

$f:\mathbb{R}\rightarrow\mathbb{R}$ be a twice differentiable function. Suppose $f(0)=0$. Prove that there exists some $\xi\in(-\frac{\pi}{2},\frac{\pi}{2})$ such that $$f''(\xi)=f(\xi)(1+2\tan^{2}{\...
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73 views

Show that the polynomial $f(x) = 3x^4 - 2 x^3 + x^2 + ax - 1$ with $a \in \mathbb{R}$ does not have all of its roots real.

Consider the polynomial: $$f(x) = 3x^4 - 2x^3 + x^2 + ax - 1$$ with $a \in \mathbb{R}$ and the roots $x_1, x_2, x_3, x_4 \in \mathbb{C}$. I have to show that the polynomial $f$ cannot have all of ...
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1answer
37 views

Find the minimum number of zeros of $g(x)=(f'(x))^2+f(x)f''(x)$in the interval $[a, e]$ for the following given information.

If $f(x)$ is twice differentiable function such that $f(a)=0, f(b)=2,f(c)=–1, f(d) =2,f(e)=0$, where $a<b<c<d<e$, then find the minimum number of zeros of $g(x)=(f'(x))^2+f(x)f''(x)$in ...
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1answer
26 views

Mean value theorem and the continuity of step length

Suppose that $f: \mathbb{R} \rightarrow \mathbb{R}, f\in C^1$. Then for two points $a,b \in \mathbb{R}$, we know by mean value theorem we know that there exists $t \in (0, 1)$ such that $$ f^\prime(a+...
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1answer
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$f:\Bbb{R}\mapsto\Bbb{R}$ and $f$ is twice differentiable.

$f:\Bbb{R}\mapsto\Bbb{R}$ and $f$ is twice differentiable such that $f(0)=2, f(1)=1, f'(0)=-2$. Prove that $\exists$ a $\varepsilon \in (0,1)$ such that $f(\varepsilon)f'(\varepsilon)+f''(\varepsilon)=...
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Using Rolle’s Theorem prove that the cubic equation $x^3+ax+b$ has at most one root

A brief explanation of Rolle’s Theorem would be appreciated! I understand that it is a special case of the mean value theorem but I am not sure how to apply it to this question. Edit: $a>0$
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Prove that $f(a)=\ell + \frac{(a-\ell)^{2}}{2}f''(c)$

Could you please help me to answer this question for freshmen students : Let $f:I=[a,b]\rightarrow\mathbb{R}$ be a function of class $2$ and suppose that there exist $\ell\in I$ such that $f(\ell)=\...
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1answer
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Why can we proof inequalities with the mean value theorem?

Can somebody explain to me WHY we can use the MVT to poof inequalities? First of all what I know: I know what the MVT says and that is pretty logical. If a function is continuous and differentiable ...
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1answer
42 views

Rolle's theorem with odd function

If I have some cubic equation $f(x)$, and I need to find how many solutions $f(x)$ has. $f'(x)$ has two zeros, does it state that $f(x)$ has $3$ solutions by Rolle's theorem? $$f(x)= x^3+2x^2-7x+1$$
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1answer
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Non negative derivative, not being zero function on all subintervals, implies strictly increasing

Suppose $ f$ is a continuous function on interval $I$ and that the derivative $ f'$ of $f$ exists and is nonnegative everywhere on the interior of $I$. Also suppose that there is no sub-interval of $...
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1answer
85 views

A function with a point of slope of exactly $2$

Let $f:[0,1]\rightarrow\mathbb{R}$ be a function continuous on $[0,1]$ and differentiable on $(0,1)$ with $f(0)=f(1)$ and $f(\alpha)=f(\beta)+1$ for some $\alpha,\beta$ such that $0<\alpha<\beta&...
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Simple questions about the error term in Lagrange polynomial interpolation.

Theorem: Let $f(x)$ be a function which is $n+1$ times differentiable over the interval $I:=\left[\min\left\{x_{i}\right\},\max\left\{x_{i}\right\}\right]$ for $0\le i\le n$ , and if $p_n(x)$ is a ...
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1answer
23 views

Prove that $g(1)=3$ if $f'(c)=2g'(c)$ for some real number $c\in(0,1)$

Let $f(x)$ and $g(x)$ be differentiable for $0\leq x\leq 1$, such that $f(0)=0,g(0)=0,f(1)=6$. Let there exists a real number $c$ in $(0,1)$ such that $f'(c)=2g'(c)$, then prove that $g(1)=3$ It is ...
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1answer
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Using Rolle's Theorem to prove that there are two roots to a function.

I am a HS student and currently learning Rolle's Theorem. I have gotten the question: Prove that there are exactly two positive real numbers $x$ such that $e^x = 3x$. This is what I have done to ...
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2answers
47 views

Suppose $f:\mathbb R \rightarrow \mathbb R$ is differentiable with $f(0)=f(1)=0$ and $\{x: f^{'}(x)=0\} \subset \{x: f(x)=0\}$.

Suppose $f:\mathbb R \rightarrow \mathbb R$ is differentiable with $f(0)=f(1)=0$ and $\{x: f^{'}(x)=0\} \subset \{x: f(x)=0\}$. Show that $f(x)=0$ for all $x \in [0,1]$. Since $f$ is differentiable ...
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1answer
33 views

A strange exercise concerning second order mean value theorem

Here is an exercise from a book: Let $a<b<c$. $f$ is twice differentiable on an open interval containing $a,b,c$. Show that there exist $d\in(a,b)$ such that $$ \frac{f(c)-f(a)}{c-a}-\frac{...
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1answer
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Mean Value Theorem demonstration

$\large f(x)=\tan^{-1}(\frac{1}{x^2}) -\ln(x^2+1)$ $\large if 1\leq x < y$ $\large \text { and } 1-\frac{2x}{1+x^2}\geq 0$ Demonstrate that $ \large |f(x)-f(y)|\leq 2|x-y|$ I managed to ...
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Prove that the second derivative has a zero

my question is: Can I use Rolle's theorem to prove that the second derivative has a zero? Consider a Real function of Real variable defined by $f(x)=(x+1)\cdot e^{x^2}$. Prove that the second ...
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3answers
56 views

Using Rolle's theorem prove that if $b^2<3ac$, then there is exactly one root to $f(x) \equiv ax^3+bx^2+cx+d=0$

Using Rolle's theorem prove that if $b^2<3ac$, then there is exactly one root to $f(x) \equiv ax^3+bx^2+cx+d=0$ I literally have no idea how to use Rolle's Theorem. Rolle's Theorem: if the ...
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3answers
112 views

Rolle's Theorem 1

I need your help. I don't know how to translate exactly the math problem I have been given for homework, since English is not my mother language, so I would really appreciate it if you didn't judge me....
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1answer
37 views

What is the maximum possible value for $f(x)$ for $x \in [0,1]$? [closed]

A function $f(x)$ is continuous and differentiable in $[0,1]$. If $f'(x) \le 10$ for all $x \in [0, 1]$ and $f(0) = 0$, what is the maximum possible value of $f(x)$ for $x$ in $[0, 1]$?
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2answers
68 views

Trying to apply Rolle's Theorem

Let $f$ be a differentiable function in the interval $[a,b]$. Prove that there exists $c\in ]a,b[$ such that $$f'(c)=f(c)\dfrac{(a+b-2c)}{(c-a)(c-b)}$$ My ideas are: maybe we need to find a ...
5
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1answer
135 views

Rolle's theorem for second derivative

A problem asks the following $f$ is a twice-differentiable function on some segment $[a,b]$ such that $f(a)=f(b)$ and $f'(a)f'(b)<0$. it asks to prove that the second derivative of $f$ vanishes ...
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1answer
51 views

Existences of Solutions by Using Rolle's Theorem

The question is: Let: $$g(x,y)=(e^x+1)y^2+2(e^{x^2}-e^{2x-1})y+(e^{-x^2}-1)$$ Then, show that there exists a constant $\bar{y}>0$, such that for any fixed $y \in [0,\bar{y}]$, the equation $...
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1answer
22 views

Relation of functions satisfying rolle's theorem

Let a > 1 and f,g: [-a,a] $\rightarrow$ R be twice differentiable functions such that for some c with 0 < c < 1 < a, f(x) = 0 only for x=-a, 0, a f$'$(x) = 0 = g(x) only x = -1, 0, 1 g$'$(...
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1answer
50 views

Prove that there exists some $c>0$ such that $f'(c)=-e^{-c}$

Let $f$ be a differentiable function on $[0,\infty)$ and $f(0)=1$. Further given that $|f(x)|\leq e^{-x}$ for all $x\geq 0$. Prove that there exists some $c>0$ such that $f'(c)=-e^{-c}$ My ...
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1answer
82 views

Proving that the following functional equation admits a solution

Given the following function: $$g(x,y) = (e^x+1)y^2+2(e^{x^2}-e^{2x-1})y+(e^{-x^2}-1) $$ I want to show that there exists a $\bar{y} \in \mathbb{R}$ such that $\bar{y} > 0$ and for a fixed $y \in ...
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1answer
69 views

Let a function f(x) be twice differentiable such that $ f(0) = 0, f(\pi/2)= 1 , f(3\pi/2)=-1$. To prove that there exists a ‘c’ in $ (0,3\pi/2)$

Let a function f(x) be twice differentiable such that $ f(0) = 0, f(\pi/2)= 1 , f(3\pi/2)=-1$. To prove that there exists a ‘c’ in $ (0,3{\pi/2})$ such that |$ f”(x) $ | is less than or equal to 1. ...
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49 views

Is this inference from Rolle's Theorem correct?

In a script I'm reading right now that is providing a proof for Cauchy's mean value theorem (extended mean value theorem), it says "We have $g(x) \neq 0$ for all $x\in ]a,b[$ (otherwise, there would ...
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2answers
103 views

Application of Rolle's theorem and determinant

$f(x) = \begin{vmatrix} \sin^3x & \sin^3 a & \sin^3 b \\ xe^x & ae^a & be^b\\ \frac{x}{1+x^2} & \frac{a}{1+a^2} & \frac{b}{1+b^2} \end{vmatrix}$ Where $0<a<b<2π$ ...
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3answers
60 views

Rolle's theorem proof in Apostol: meaningfulness of interior

According to the statement of the Rolle's theorem in Apostol calculus 1, we need to have a continuous function on $S = [a, b]$, and this function should have a derivative on the interior of $S$. I do ...
3
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4answers
104 views

Some property of differentiable function

Let $f:[0,2]\to\mathbb{R}$ be a continuous function and $f$ is differentiable on $(0,2)$, and let $f(0)=f(2)=0$. Now, suppose that there is a point $c\in(0,2)$ such that $f(c)=1$. Then, there is a ...
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2answers
87 views

Spivak, Calculus 3rd Ed. Chapter 11 problem 32

I made it this far through the book but now I'm really stumped. Here's the problem: Suppose that $f$ and $g$ are two differentiable functions which satisfy $fg'-f'g=0$. Prove that if $a$ and $b$ ...
3
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3answers
158 views

Given $f(x) = x \sin\frac1x$, find roots of $f'(x)$ in the interval $0\le x \le \frac 1{\pi}$.

This question is taken from book: Advanced Calculus: An Introduction to Classical Analysis, by Louis Brand. The book is concerned with introductory analysis. If $f(x) = x \sin\frac1x\;(x\ne 0), f(0)...
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1answer
57 views

A function $g(x)$ has one and only one real root if $g'(x)\leq k <0$.

$g : \mathbb{R} \to \mathbb{R}$ is differentiable on $\mathbb{R}$. Then $g(x)$ has one and only one real root if $g'(x)\leq k <0$. Proof attempt: Let us assume the contrary, i.e. $g(x)$ has ...
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1answer
39 views

Rolle's and Lagrange's theorems in parametric equations

I was trying to solve some calculus problems and I came across with some doubts related to two of them. Given the parametric equation $$ (x,y)=(3-3\cos^2 (t),3-3\cos(t)\sin(t)) \quad 4 \pi /3 < t ...
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3answers
125 views

Root multiplicity for non-polynomial function

I know that if $f$ is a polynomial and $f(a)=0,f'(a)=0...,f^{(k)}(a)\neq 0$, then $a$ is a root of multiplicity $k$. Does this work for a differentiable function that is not a polynomial? I have seen ...
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1answer
74 views

Geometry of mean value theorem? [duplicate]

Usually the mean value theorems (including the general one) is extended from the rolle theorem by introducing a function. Does this function has anything to do with geometric transformation? Can ...