Mathematics Stack Exchange
Mathematics Stack Exchange

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.

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.

283 questions
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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 root in [-2,2] (4) no root in [-2,2] I am doing hit…
Samar Imam Zaidi
  • 8,800
2
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2 answers

Let $f(x)=\begin{cases} x^a\ln x & x>0 \\ 0 & x=0 \end{cases}$. Find the value of $a$ for which Rolle’s theorem is applicable in $[0,1]$

$$f(0) = f(1)=0$$ So according to Rolle’s theorem, there exists a number $c \in (0,1)$ such that $f’(c) =0$ $$f’(x)=x^{a-1} + ax^{a-1} \ln x$$ $$f’(x)=x^{a-1} (1+a\ln x)=0$$ $$a=\frac{-1}{\ln x}$$ How o I find the value of $a$ from here?
Aditya
  • 6,191
1
vote
1 answer

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 have two solution $x_1$ e $x_2$ such as…
Michele Assirelli
  • 11
1
vote
1 answer

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 answer: $f(0) = 1 > 0$, $f(1) = e - 3 < 0$. There must…
multipledifferentones
  • 37
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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 $f'(\xi)=f(\xi)$.
Wang Aliber
  • 41
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A problem on construction of a function

Given that $f(x)$ is continuous on $[a,b]$ and differentiable on $(a,b)$, it is to be proved that there exists an $\xi \in (a,b)$ such that $$f'(\xi)=\dfrac{2}{a-\xi}\cdot (f(\xi)-f(b)).$$
zisheng
  • 3
0
votes
2 answers

Let $f: [0,6] \rightarrow \mathbb R$ be a three times differentiable function such that $f(0)=f(1.4)=f(3.9)=f(5.2)$.

Let $f: [0,6] \rightarrow \mathbb R$ be a three times differentiable function such that $f(0)=f(1.4)=f(3.9)=f(5.2)$. Prove that there is c $\epsilon$ (0,6) such that $f^{'''}(c)=0$. Now using Rolle Theorem I know that from (0,5.2) I have that…
Nuemann12
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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? $$f(x)= x^3+2x^2-7x+1$$
Veronika Kovaleva
  • 23
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1 answer

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 get a quadratic équation in term of $c$ through the…
SAM.Am
  • 387
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3 answers

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. Function $f:[0, \pi] \rightarrow \Bbb R$ is…
calliope
  • 3
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votes
1 answer

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. my conjecture is that question is wrong. by given…
maveric
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Rolle's Theorem Function

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.
Peter Ferri
  • 49
0
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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 relationship between $a$ and $b$ is $b = a(1 - e^2)$ the…
pi314
  • 11
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$P_n(x):={1\over{2^n{n!}}}{d^n\over dx^n}[(x^2-1)^n]$ has N Distinct Root

Claim Let $P_n: \Bbb R \mapsto \Bbb R $, $n \in \Bbb N\cup\{0\}$ $P_n(x):={1\over{2^n{n!}}}{d^n\over dx^n}[(x^2-1)^n]$, then $P_n$ has n distinct roots in ]-1, 1[ Proof $1$. Base When $n=1$, $P_1(x) = {1 \over 2}{d \over dx}[x^2-1]=x$ and…
Beverlie
  • 2,645
-1
votes
3 answers

Problem with Rolle's theorem

I really don't understand how to solve this. I have tried some solutions but they all assume that $f(a)=f(b)$. Can you give me more a more substantial hint? Here is the problem
Aldan D-Akira
  • 3