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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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Constructing a new function $F(x)$ and apply Rolle's Theorem

I have tried to construct a new function $F(x)$ such that $F(a)=F(b)=0$ and apply Rolle's Theorem, but the question is: there's two variables $f(c)$ and $g(c)$ on the rate of change (left-hand side) ...
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106 views

Apply Rolle's theorem to find real roots

Suppose the function $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$ such that $f(a)=f(b)=0$. Prove that there exist a point $c\in(a,b)$ such that $$f(c)-f'(c)=0$$ From the question ...
3
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2answers
73 views

The equation $ f'(x)=f(x)$ admits a solution

let $f :[0,1]→\mathbb R$ be a fixed continous function such that f is differentiable on (0,1) and $ f(0)=f(1)=0$ .then the equation $ f'(x)=f(x)$ admits 1.No solution $x\in (0,1)$ 2. More than ...
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3answers
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Application of mean value and Rolle's theorems - twice differentiable functions

Let $f$ be a function from $[a,b]$ to $\Bbb{R}$ that is twice-differentiable (that is, $f'$ and $f''$ exist), and assume that $f(a) = f(b) = 0$ and $f''(x) \leq 0$ for every $x\in (a,b)$. Show that $f(...
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3answers
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Prove, without using Rolle's theorem, that a polynomial $f$ with $f'(a) = 0 = f'(b)$ for some $a < b$, has at most one root

Prove the following without using Rolle's Theorem: If $f$ is a polynomial, $f'(a) = 0 = f'(b)$ for some $a < b$, and there is no $c \in (a,b)$ such that $f'(c) = 0$, then there is at most one ...
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1answer
57 views

If $f(x)$ has $n$ distinct roots in $R$, then $f'(x)$ has $n-1$ distinct roots in $R$ Without Rolle's Theorem

Prove that: If $f(x)$ has $n$ distinct roots in $R$, then $f'(x)$ has $n-1$ distinct roots in $R$ Without Rolle's Theorem. I know in this topic, There is proof with Rolle's theorem. It uses that if ...
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25 views

if $f'(x)=k$ then $f(x)=kx+q$ for some $q$

Here's a question from my analysis textbook from the section of Mean Value Theorem: Suppose $f$ is continuous on $[a,b]$ , differentiable on $(a,b)$ and that $f'(x)=k$ for all $x\in(a,b)$ for ...
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0answers
33 views

Proving that first derivative vanishes

Let $f$ be a real function which is differentiable on some interval $[0,a]$ ($a>0$) that satisfies : $f'$ is continuous on $[0,a]$ $f'(0)=f(0)=0$ $f(a).f'(a)<0$ It is asked to prove that $f'$ ...
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2answers
56 views

Proving $f(t)=f(0)$ using Mean Value Theorem

Suppose that function $f: [0, \infty) \to \mathbb{R}$ is differentiable at every $t > 0$ and continuous at $= 0$. Show that if $f'(t) = 0$ for all $t > 0$ then $f'(t) = f'(0)$ for all $t ≥ 0$. ...
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1answer
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Rolle's Theorem and vertical tangents

Consider the function $f(x)=\sqrt[3]{x-2}-x+4$ on $[1,3]$. $f$ is certainly continuous on $[1,3]$ and $f(1)=2=f(3)$, but $f$ is not differentiable on $(1,3)$ since $$f^\prime(x)=\frac{1}{3\sqrt[3]{(x-...
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1answer
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The intermediate value property

I want to prove this statement, Assume $f:(a,b)\to \mathbb R$ has intermediate value property, then $f$ cannot have jump discontinuities. So, i have two way to prove; assume $f:(a,b)\to \mathbb R$ ...
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1answer
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Can we conclude that $f$ is continuous on the interval $[a,b]$?

Corollary $1$. If $f$ is differentiable on the interval $a<x<b$, then the zeros of $f$ are separated by zeros of $f'$. proof. Let $x_1,x_2,...,x_n$ be the real roots of the function $f$. ...
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1answer
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Rolle's Theorem Function [closed]

Find all numbers, $c$ that satisfies the conclusion of Rolle's Theorem for the following function, $f(x)=x^2−10x+10,[0,10]$ I haven't learned this theorem yet and am confused on what to do.
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2answers
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Rolles theorem used for solving equation $ax^3+bx^2+cx+d=0$

If a,b,c,d are Real number such that $\frac{3a+2b}{c+d}+\frac{3}{2}=0$. Then the equation $ax^3+bx^2+cx+d=0$ has (1) at least one root in [-2,0] (2) at least one root in [0,2] (3) at least two ...
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4answers
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If Rolle's Theorem is assumed to be true, doesn't that prove the MVT?

If we assume that Rolle's Theorem is true is it practical to say that it also proves the MVT? My reasoning is that even though Rolle's Theorem is the special case for when $f(a)=f(b)$ and the secant ...
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0answers
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How to use mean value theorem to derive this Taylor expansion equation?

The mean value theorem says there exists a $c$ in $(a, b)$, such that $f(b)-f(a)=f'(c)(b-a)$. There should also be equation for second order, $f(b)-f(a)=f'(a)(b-a)+1/2*f''(d)(b-a)$ where $d$ is ...
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1answer
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Prove that a function has a root in interval, when you know, by Rolle, there is a root of the derivative, but you don't know it.

Let $a_1,\ a_2,\ b_1,\ b_2 \in \mathbb{R}$. Prove that: $$a_1\cos(x) + a_2\sin(x) + b_1\cos(2x)+b_2\sin(2x)=0$$ has at least one root in $(0,2\pi)$. I started by creating a function: $f:[0,2\pi]\...
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1answer
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Analysis: Differentiation problem

I have been spending times on the following question for 2hrs, and I couldn't get any kind of proper proof yet. Q: Let $f$ be continuous on $[0,\infty)$ and differentiable on $(0,\infty)$. If $f(0)=...
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2answers
108 views

Rolle's theorem on $e^x\sin x-1=0$

Prove that between any two real roots of the equation $e^x\sin x-1=0$ the equation $e^x\cos x+1=0$ has at least one root. My attempts: By Rolle's theorem, the derivative of $e^x\sin x-1=0$ has at ...
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Showing point in f(x) such that f'(x)=0 exists [closed]

let $f(x)=x^4 + \sin x$ Show that there exists $x \in (-2,2)$ such that $f'(x) = 0$.
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How to use Rolle's Theorem to show that $f$ has at most one fixed point? NOT DUPLICATE [duplicate]

Let $a,b \in \mathbb{R}$ be such that $a\lt b $. Suppose that $f:[a,b]→\mathbb{R}$ is a continuous function on $[a,b]$ and is differentiable on $(a,b)$ and that $f'(x) \gt 1$ , for all $x\in(a,b)$. ...
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3answers
151 views

Presenting a lecture on Mean value theorems.

I've to give a model lecture on Mean Value theorems(Rolle's theorem,Lagrange's Mean value theorem,Cauchy's Mean value theorem).But i do not know what content i should include in the lesson(I do not ...
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487 views

Rolle's theorem: what's the right statement of the theorem?

In the fourth edition of "Introduction to Real Analysis" by Bartle and Sherbert, theorem 6.2.3 (Rolle's theorem) states, Suppose that f is continuous on a closed interval $I := [a, b]$, that the ...
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2answers
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Application of Derivatives (Rolle's Theorem?)

Let there be a cubic equation $f(x)=ax³+bx²+cx+d=0$, and the coefficients of the equation be related by $-a+b-c+d=3$ and $8a+4b+2c+d=6$. How can I show that the quadratic equation $3ax²+2bx+c-1$ has a ...
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Rolle's theorem without compact domain hypothesis [duplicate]

Good evening everyone, I'm asking for a proof of "Rolle's theorem generalization". The thesis is as follows: Let be $a \in \mathbb{R}$ and $f:[a,\infty) \longrightarrow \mathbb{R}$ a continuous ...
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1answer
51 views

Prove by Taylor expansion or mean value theorem

If a particle moving on the Euclidean line traverses distance $1$ in time $1$ starting and ending at rest, then at some time $t \in [0, 1]$, the absolute value of its acceleration should be at least $...
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1answer
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Related rate problem (absolute value of its acceleration should be at least 4.) [closed]

If a particle moving on the Euclidean line traverses distance 1 in time 1 starting and ending at rest, then at some time t ∈ [0, 1], the absolute value of its acceleration should be at least 4. So,...
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1answer
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Show that $\exists c\in (a,b)$ such that $2f(c)+f'(c)=0$ [duplicate]

Let $f$ be a function continuous on $[a,b]$ and differentiable on $(a,b)$, $f(a) = f(b) = 0$ Show that $\exists c\in(a,b)$ such that $2f(c)+f'(c) = 0$ This is a problem on continuity and ...
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3answers
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Using the Mean Value Theorem, show that for all positive integers $n$, $n\ln(1+\frac{1}{n}) \leq 1$

I am not too familiar with Rolle's Theorem and MVT so this question is a little bit tricky for me. I tried it by letting some $$f(x)=x\ln(1+\frac{1}{x})-1.$$ Then by MVT, in the interval $x \in [0,n]$...
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2answers
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Can $a$ and $b$ in Rolle's Theorem use Infinities?

I am actually analyzing the behavior of the function $$f(x)=\frac{x}{1+x^2}$$ we have $f$ is Continuous and Differentiable over $\mathbb{R}$ Also $$\lim_{x \to -\infty} f(x)=\lim_ {x \to \infty}f(...
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2answers
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Show that a function is surjective using Rolle's Theorem

Let a,b ∈ $\mathbb{R}$, a < b and g: [a,b] → $\mathbb{R}$ is a continuous function, differentiable in ]a,b[, such that g(a) = g(b) = 0 and g(x) $\neq$ 0 $\forall$ x ∈ ]a,b[. Show that the ...
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4answers
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Can we say there is exactly one root of $x^4-7x^3+9=0$ in $(1, \:\: 2)$?

We have $f(1) \gt 0$ and $f(2) \lt 0$. Hence, the intermediate value theorem (IVT) guarantees at least one root in $(1, \: \: 2)$. Now, let's assume there are two roots $\alpha$ and $\beta$ in $(1, \:...
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1answer
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Why Rolle's theorem is giving two roots here?

I am trying to find Number of distinct roots of $$f(x)=x+5\cos x=0$$ in $\left[0, \pi \right]$ we have $f(0)=5 \gt 0$ and $f(\pi)=\pi-5 \lt 0$ so by IVT we have at least one root in $\left[0, \pi \...
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1answer
40 views

$x-\frac{x^2}{2}<\ln(1+x)<x-\frac{x^2}{2(1+x)}, x>0$ [duplicate]

Show that- $$x-\frac{x^2}{2}<\ln(1+x)<x-\frac{x^2}{2(1+x)}, x>0$$ I can prove this by observing that at $x=0$, the three functions in the above inequality (say $f_1<f_2<f_3$) are zero. ...
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1answer
145 views

Maximum Uniqueness and Rolle's Theorem

Consider the function $y(x)=f(x)(1-x)$ where $x\in[0,1]$, $y(0)=y(1)=0$. Knowing that $f(x)$ is continuous differentiable everywhere, $f(0)=0$, $f(1)=c$ where $0<c<1$, $f'(x)>0$, can one ...
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1answer
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Show that a non-linear equation has exactly two roots

How can I show, that the following equation $xe^{-x} = e^{-3}$ has exactly two roots? I found here an answer, but it uses many other theorems as well, and we have not yet studied so advanced concepts....
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3answers
37 views

Determinate $\lambda\in R$ so that the following equation has 2 real,distinct solutions.

Determinate $\lambda\in R$ so that the following equation has 2 real,distinct solutions. $$2x+\ln x-\lambda(x-\ln x)=0$$ I think this should be solved using Rolle property for finding intervals with ...
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0answers
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Find the interval for application of Rolles theorem.

$x^3+x^2+m=5\ln{|x|}$, depending on any real $m$. I did the first steps and did the derivative, got the solutions from the derivative, $x=1$; $f(1)=2+m$ and got some solutions but the answer says ...
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2answers
67 views

How to prove that $\cos x< \cos (\sin x)$ using the mean value theorem?

Using the mean value theorem, prove that, for $0<x<\pi/2$, $\cos x<\cos(\sin x)$. I was trying to use the mean value theorem but I got lost. I am a newbie to this please explain this as you'...
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1answer
55 views

Verify the Rolle's theorem when $f(x)=(x+1)^m(x-1)^n,\;\;-1\leq x\leq 1$

Verify the Rolle's theorem when $f(x)=(x+1)^m(x-1)^n,\;\;-1\leq x\leq 1$ and show this result is not true for $f(x)=2x^{-2},\;\;\text{on}\;[-1,1]$ and $g(x)=|x|,\;\;\text{on}\;[-1,1].$ I know that ...
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2answers
34 views

Finding a function that fits this description (Rolle's Theorem)

Context In the previous question, we were asked to prove Rolle's Theorem: If f is continuous on (a,b) and f (a) = f (b) = 0 then f (c) = 0 for a certain c in (a,b). Question In the case of I = [0, ...
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1answer
58 views

Use induction to show that a polynomial with $k$ terms has $k$ solutions.

First Part of the problem: Let $n_1 < . . . < n_k$ be non-negative integers, let $a_1, . . . , a_k$ be positive real numbers and let $f(x) = a_1x^ {n_1} + · · · + a_kx^{n_k}$ . Suppose $0 < s ...
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1answer
105 views

Proving Rolle's Theorem

Trying to prove Rolle's Theorem, which says that for a function $f$ continuous over $[a,b]$ and differentiable over $(a,b)$ (no idea why the endpoints aren't included here), such that $f(a) = f(b)$, ...
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1answer
2k views

Showing a function has exactly one zero with IVT and Rolle's Theorem

This is an exercise that appears on differential calculus exams at my university. I'm typing up a thorough response to this exercise here to share with my class, and maybe it'll help other students ...
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2answers
54 views

Proving a polynomial has x amount of zeros

I am new to this thread so sorry if I violate any rules or whatever, but anyway in Calculus right now we are doing stuff related to Fermat's Theorem, Rolle's Theorem, and Intermediate value theorem. I ...
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1answer
66 views

The equation $x^3-6x^2-5x+12=0$ has at least one root between :-

The options to this question are $(5,6) ; (0,1) ; (1,2)$ and $(2,5)$. My attempt : I assumed the polynomial function to be f'(x) and found f(x) by integrating it. Now, if I am able to find two ...
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1answer
33 views

Rolles theorem word problem - find the c - double checking my answer

Problem: Find the relationship of $a$ and $b$ so that Rolles theorem applies for the function $f(x) = ax^2 + b(\ln x)$ on $[1,e]$. Find the value of $c$ for which it is verified. Answer: the ...
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7answers
2k views

Prove that a degree-$6$ polynomial has exactly $2$ real roots

I have the function $f(x)={7x^6+8x+2}$ and I'm trying to prove that $f$ has exactly 2 real roots. What I've done: The only kind of solution I have come up with is essentially guessing pairs of ...
0
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3answers
71 views

Determine all $x \in \mathbb{R}$ so that $2^x = x^2+1$

I am studying for my exam in two weeks and currently going over an old exam where I found the following task: Determine all $x \in \mathbb{R}$ so that $2^x = x^2+1$. The hint is: Determine the first ...
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2answers
40 views

Number of distinct real roots of derivative of a function

Let we have a function: $f(x)=(x-1)(x-2)(x-3)(x-4)(x-5)$ The question is: There are how many distinct real roots of the polynomial $\frac{d}{dx}f(x)=0$ has?. My approach: Clearly, $f(x)=0$ has ...