Questions tagged [mean-value-theorem]

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Generalization of mean value theorem?

For differentiable function $f: \mathbb{R} \to \mathbb{R}$ the mean value theorem states: given $a, b$ there exists $c$ such that $$ f(b) - f(a) = f'(c) (b-a). $$ Is there something similar for $f: \...
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Riemann Integration and Supremum

Suppose we have $f(t, \mathbf{s}): \mathbb{R} \times \mathbb{R}^{g}$ where $\mathbf{s} \in [0, 1]^{g}$ with $g \in \mathbb{Z}_{+} \cup \infty$, $t \in \mathbf{t}$ where $\mathbf{t}$ is a set of tagged ...
SiegAndy's user avatar
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1 answer
71 views

How can I evaluate the limit of $\theta$ in the expression $f(x+h)-f(x)=h f'(x+\theta h)$?

I'm working on a problem which asks to calculate the value of $\lim_{h\rightarrow 0} \theta$,where $\theta$ comes from the mean value theorem $f(x+h)-f(x)=hf'(x+\theta h)$,and $f$ is first order ...
Egyptian's user avatar
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Mean value theorem $f'(a)=f'(b)$ [duplicate]

Let $f$ be differentiable function such that for some $a<b$ we have $f'(a)=f'(b)$. Prove that there exists $x \in (a,b)$ such that $f(x)-f(a)=f'(x)(x-a)$. I've already proved that without loss of ...
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2 answers
148 views

Multidimensional Mean Value Theorem with arbitrary norm

In the question Multivariate Mean Value Theorem Reference was written the following statement for $x,y\in \mathbb{R}^{n}$ \begin{equation} ||f(x) - f(y)||_q \leq \sup_{z\in[x,y]}||f'(z)||_{(q,p)}||x-...
Иван Петров's user avatar
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$\alpha$-Hölder function with $\alpha > 1$ is constant.

Let $\alpha > 1$ and $c \in \mathbb{R}_{+}$. If $f: U \rightarrow \mathbb{R}^{n}$, where $U \subseteq \mathbb{R}^{m}$ is an open set, satisfies $|f(x) - f(y)| \leq c|x-y|^{\alpha}$ for any $x,y \in ...
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What is the Second mean value theorem for integrals?

I was attempting to solve this limit $$\lim_{n \to \infty}\int_{0}^ \infty \frac{nx \arctan(x)}{(1+x)(n^2+x^2)}dx $$ After some time I gave up and saw the solution. The solution involves the Second ...
pie's user avatar
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1 answer
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Prove that $\lim_{x\rightarrow\infty}f(2x)-f(x)=0$ using the Mean value theorem

I encountered the following question, regarding the Mean value theorem: Let $f$ be some function such that it is differentiable the interval $(5, \infty)$, and satisfies $\lim_{x\rightarrow\infty}f'(x)...
Dani's user avatar
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How does the mean value theorem with Hölder seminorms depend on dimension?

I am reading through Krylov's Lectures on Elliptic and Parabolic Equations in Hölder Spaces. In an attempt to prove the interpolation inequalities for parabolic PDEs, I've stepped back in the text to ...
IdenticallyEulerian's user avatar
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Showing that $f$ is the zero map by applying the mean value theorem [duplicate]

Let $f:[0,1] \to \mathbb{R}$ be a differentiable function such that $f(0) = 0$ and $|f'(x)| \leq |f(x)|$ for all $x \in [0, 1]$. Show that $f(x) = 0$ for all $x \in [0, 1]$. My attempt is this: ...
MrGran's user avatar
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Mean value theorem for second derivative, Zorich [duplicate]

I have the following problem (Zorich 5.3.7(b)): let $f \in C(I)$, $f$ have derivative on $I$ (so $f'$ exists on $I$, but might be discontinuous in general) $[a, b] \subset I$ and $f \in C^{(2)}((a, b))...
brokoner12's user avatar
2 votes
1 answer
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Limit of a sequence of the real root of a family of polynomials

Consider a sequence of polynomials $P(x)=x^n+x^{n-1}+x^{n-2}+...+x^2+x-1, n>2$. (i) Prove that it has a unique positive real root $x_n$ (ii) Find $$\lim_{n \to ∞} 2^n (x_n - 1/2)$$ The first part ...
Cognoscenti's user avatar
3 votes
2 answers
76 views

Existence of $1/(e-1)$ value of a continous and differentiable function $f:[0,1]\to \mathbb R$ given $f(1)=1, f(0)=0$

Let $f:[0,1]\to \mathbb R$ be a continous function on $[0,1]$ and differentiable on $(0,1)$, $f(0)=0, f(1)=1$. Prove that there exists $c\in (0,1)$ so that $f(c)+\frac{1}{e-1}=f'(c)$ where $e$ is the ...
user1259172's user avatar
2 votes
1 answer
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Question about theorem 9.21 in Rudin's PMA

The theorem in question is: Now, the part that I have a problem is the following highlighted text: Theorem (5.10) is I don't understand what function plays the role of $f$ at (5.10). What i've ...
Manuel Ocaña's user avatar
1 vote
2 answers
89 views

Use the MVT for Integrals to bound $\int_0^1\frac{x^6}{\sqrt{1+x^2}}dx$

I have an exercise to use the mean value theorem for integrals to show that $$\frac{1}{7\sqrt 2}\le\int_0^1\frac{x^6}{\sqrt{1+x^2}}\ dx\le\frac{1}7$$ I've determined that the integrand is increasing ...
Addem's user avatar
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Find sufficient condition for having $f(x)>0$

I have to find a sufficient condition for $f(0)$ for having $f(x)>0$, if $f:[0,3]\to\mathbb{R}$ continuous and differentiable, and $f'(x)>-1$ for $x\in(0,3)$. Since $f$ is differentiable in (0,3)...
axi's user avatar
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A Detail in Mean Value Theorem for Integrals

Here is the most common and basic version of the Mean Value Theorem for Integrals: Let $f$ be continuous on $[a, b]$. Then there exists $c \in [a, b]$ such that $f(c) = \frac{1}{b-a} \int_a^b f(x) dx$...
Joseph's user avatar
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$f(U)$ is an interval of length $\leq 3m$.

Let $U = \{x \in \mathbb{R}^{m}; |x_i| < 1, i = 1, \ldots, m\}$ and $f: U \rightarrow \mathbb{R}$ a differentiable function such that $|\partial_{x_i} f(x)| \leq 3$ for all $x \in U$. Then, $f(U)$ ...
huh's user avatar
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Counter example: continuity assumption of mean value theorem [duplicate]

Let's consider $f:[a,b]\to\mathbb{R}$ which is differentiable in each $x\in ~\!]a,b[$. Consequently $f$ is also continuous in those points. But we don't require that $f$ is continuous in $a$ and $b$. (...
Philipp's user avatar
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Taylor expansion to reformulate difference of function at two points

I'm trying to understand the proof of Proposition 1.13 in Tsybakov's book Introduction to Nonparametric Estimation. I'm stuck on the following detail. Let $f : \mathbb{R} \mapsto \mathbb{R}$ be $\ell$ ...
digbyterrell's user avatar
4 votes
1 answer
132 views

Integral inequality : for $f\in C^{4}[0,1]$ with $f(0)=f(1)=f^\prime(0)=f^\prime(1)=0$, show that $\int_0^1|\frac{f^{(4)}(x)}{f(x)}|dx\geq 192$

I was interested in the problem stated below: Problem 0. $f\in C^{4}[0,1]$ with $f(0)=f(1)=f^\prime(0)=f^\prime(1)=0$, show that $\int_0^1|\frac{f^{(4)}(x)}{f(x)}|dx\geq 192$ This problem is assigned ...
CCQ's user avatar
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$f(x)$ has 3 distinct solutions on $[a,b]$ and $f(x)$ has a continuous second derivative on $[a,b]$, then $f''(x)+2f(x)$ has solutions on $[a,b]$ [closed]

Prove if $f(x)$ has 3 distinct solutions on $[a,b]$ and $f(x)$ has a continuous second derivative on $[a,b]$, then $f''(x)+2f(x)$ has solutions on $[a,b]$. I'm stuck for ideas on this problem. Where ...
Mark's user avatar
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0 answers
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On connected sets of $\mathbb{R}^{n}$.

Let $U \subseteq \mathbb{R}^{n}$ be an open and connected set. If $f: U \rightarrow \mathbb{R}$ has partial derivatives equal to zero on every point of $U$, then $f$ is constant. My idea to prove the ...
user avatar
-1 votes
1 answer
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Which function is differentiable on $[a,b]$ but doesn't reach an extreme on $[a,b]$? [closed]

I think this is an old question but I can't find any evidence. When I learn about Flett's theorem and Darboux's theorem, we consider it to have an extrema on $(a,b)$ and on two boundaries. But in case ...
Mark's user avatar
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2 votes
1 answer
110 views

Alternative solution for integral limit using mean theorem

Let $$ I_{n} = \int_{0}^{1}\frac{x^{n}}{x^{n}+1}dx $$ I would like to know whether it's possible to prove that $$ \lim_{n \to \infty} I_{n} = 0 $$ using the mean theorem for integrals. It is quite ...
Andrei's user avatar
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For a differentiable function $f$ , does this mean value theorem iteration method converge to the root of $f?$

Suppose $a<b,\quad f:(a,b)\to\mathbb{R}$ is a differentiable, nowhere-linear function$,\ f(c)=0 $ for some $c\in (a,b),\quad$ and $f'(x)\neq 0\ \forall\ x\in (a,b).\ $ Let $x_0=a;\ x_1=b.$ Nowhere-...
Adam Rubinson's user avatar
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1 answer
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How to prove that a 2D function f(x,y) is always equal to a 1D function g(x) if f(x,y) always has zero gradient in the y direction

Prove that if $f(x,y)$ is continuously differentiable in the whole plane, and if $f_y\equiv0$, then $f(x,y) = g(x)$, where g is a function of one variable. [Hint: let $g(x) = f(x,0)$ and apply the ...
gnitsuk's user avatar
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2 votes
2 answers
147 views

Let $f$ be continuous in $[0,2]$ and differentiable in $(0,2)$ such that $|f'(x)|\leqslant1$ and $f(0)=1=f(2)$. Prove that $f(x)\geqslant0$.

Let $f$ be a continuous function in $[0,2]$ and differentiable in $(0,2)$, such that $|f'(x)|\leqslant1$ for all $x\in(0,2)$ and also $f(0)=1=f(2)$. Prove that $f(x)\geqslant0$ for all $x\in(0,2)$. My ...
MATH14's user avatar
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1 answer
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Derivative Function

Given $f(x) \ge 0$ for every $x \in (a,b)$ and $c \in (a,b)$. If $f'(c)$ exist and $f(c) = 0$, proof $$f'(c) = 0.$$ My Attempt: By using Mean Value Theorem: $$f'(c) = \frac{|f(b)-f(a)|}{|b-a|} \ge 0.$$...
Niccolo's user avatar
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2 answers
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Find functions $f(x)$ and $g(x)$ such that the following conditions are satisfied for all $x > 0$: [closed]

$0 < f(x) < 1$, $g(x) < \frac{f(x)}{x} < c$ for some constant $c$, $\frac{d}{dx}g(x) > 0$.
Pankaj Mishra's user avatar
2 votes
2 answers
182 views

If $\theta(h)$ is such that $f(3+h)-f(3) = hf'(3+\theta(h)h)$, what is $\lim_{h \to 0} θ(h)$?

Problem. Let $f(x) = \sqrt[3]{x}$ for $x\in(0,\infty)$, and $\theta(h)$ be a function such that $$f(3+h)-f(3) = hf'(3+\theta(h)h)$$ for all $h \in (−1, 1)$. Then $\lim_{h \to 0} θ(h)$ is equal to? (...
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6 votes
3 answers
256 views

With some conditions on $f$, resolve the claim $|f(x)-f(y)|\le|x-y|$ for all $x,y\in[0,1]$.

Here's the problem . . . Prove or disprove:$\;$If $f:[0,1]\to\mathbb{R}$ is a twice differentiable function such that $f(0)=f(1)=0$. $|f''(x)|\le 2$ for all $x\in[0,1]$. then $|f(x)-f(y)|\le|x-y|$ ...
quasi's user avatar
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Ratio of a function and its integral inequalities

Consider monotone decreasing functions $f,g:[0,\infty)\longrightarrow[0,1]$, such that $f\le g$, $f(0)=g(0)=1$, both vanishing as $x\longrightarrow\infty$ and integrable. I would like to compare $R_f(...
xyz's user avatar
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3 votes
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Proof of chain rule using mean value theorem

Is this proof of the chain rule valid ? We want to find the value of : $$(g\circ f)'(x) =\lim\limits_{h \to ~0}\frac{(g\circ f)(x + h) - (g\circ f)(h)}{h} \tag{1}$$ But we know from the mean value ...
aidaGoG's user avatar
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2 votes
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Question regarding Mean value theorem and a monotonically increasing function $f'$

I was given the following information about a function: Let $f$: [0,$\infty$) be a function so that $f(0)=0$ $f$ is continuous on [0,$\infty$) $f$ is differentiable on (0,$\infty$) the function $f'$: ...
Muhammad Abdurrahman Ullah's user avatar
2 votes
0 answers
99 views

Let $f$ be continuous on $[a,b]$ and differentiable on $(a,b)$. If $\lim_{x\to{a}} f^\prime(x)$ exists, show that $f$ is differentiable at $a$ [duplicate]

I have some troubles in trying to prove the following result and possibly need some guidance: Let $f$ be continuous on $[a,b]$ and differentiable on $(a,b)$. If $\lim_{x\to{a}} f^\prime(x)$ exists, ...
Muhammad Abdurrahman Ullah's user avatar
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0 answers
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MVT Problem : Prove $f^{'}(a)=\lim_{x\to{a}} f^{'}(x)$

I'm trying to solve a problem that goes as follows : Let $f$ be continuous on $[a,b]$ and differentiable on $(a,b)$. If $\lim_{x\to{a}} f^{'}(x)$ exists, show that $(1)$, $f$ is differentiable at $a$,...
aort01's user avatar
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1 answer
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Bounded second derivative also bounds the function

I've been struggling with this problem for a few days: Let $f:\mathbb{R}^2\rightarrow \mathbb{R}$ defined as $f(x,y)=xg(y)-yg(x)$, where $g:\mathbb{R}\rightarrow \mathbb{R}$ is such that $g\in C^2$, $...
Arthur's user avatar
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2 votes
0 answers
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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 ...
CESTU's user avatar
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0 answers
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If we know a bound for the third derivative and 3 roots for a function, how can we derive its possible maximum value?

The following problem was presented by a guest who gave a talk at our problem-solving seminar: Suppose $f$ is at least $3$ times differentible on $(-1,1)$, and $f,f^{'}, f^{''}, f^{'''}$ are ...
StAKmod's user avatar
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1 vote
1 answer
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Finding a constant for the mean value theorem

If $ g(x) \geq 0 $ and, I want to calculate $ \int f(x)g(x) dx $ for some $ f(x)$. Can I find a constant $c$ satisfying $$ \int f(x)g(x) dx = \int cf(x) dx \text{ ? }$$ or even can I claim that $ c= ...
Lee's user avatar
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Prove using Lagrange's Mean Value Theorem, $\frac{x^2}{2} > x - \ln(1 + x) > \frac{x^2}{2(1 + x)}$ for all $x>1$

By taking $f(x) = \ln(1 + x)$ and using the inequality $\frac{1}{1 + x} < \frac{1}{1 + c} < 1$ for $c \in (0, x)$, applying LMVT gives $\frac{x}{1 + x} < \ln(1 + x) < x$. By taking $f(x) = ...
112120's user avatar
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0 answers
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How to understand "as $x$ tends to $a$, value $c$ also tends to $a$" at the proof L'Hospital's Rule using Cauchy theorem?

When I read the proof of L'Hospital's Rule, the exact meaning of "x tends to a" made me confused. I'll write briefly about the proof: $\lim_{x \to a^+}f(x) = \lim_{x \to a^+}g(x) = 0$ and $...
王雨阳's user avatar
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41 views

Mean value theorem and Lp functions

Let $u, v \in L^p(\Omega)$ where $\Omega\subset \mathbb{R}^d$, and $S:\mathbb{R}\rightarrow \mathbb{R}$ is the sigmoid function, $$S(x) = \frac{1}{1+e^{-x}}.$$ I want to know whether I can apply mean ...
dk_'s user avatar
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1 vote
0 answers
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How to use the Mean Value Theorem to "establish bounds"?

I am faced with the following question: Use the mean value theorem to establish bounds in the following cases. (a) For $-\ln(1-y)$, by considering $\ln x$ in the range $0<1-y<x<1$. (b) For $...
Kalo's user avatar
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0 answers
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A Particular Version of The Lagrange Mean Value Theorem for Complex Valued Functions

I am given the following homework question to work with: I am done with parts (a) and (b). But I cannot work on other parts because the concepts are not clear to me: what is exactly meant by "...
autodidacti's user avatar
1 vote
0 answers
43 views

MVT for integrals, proof attempt

I know the proof of the theorem by applying the mean value theorem, but the teacher asked us to prove it by integration by parts. However, I insist that it's not possible, because integration by parts ...
Daniel Amaya Zabala's user avatar
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0 answers
71 views

A challenging problem in differential calculus that cannot be solved using conventional methods. [duplicate]

The function f(x) is differentiable on [a, b] and has a third derivative on (a, b).Prove that:there is a $\xi \in(a, b)$,such that $$\begin{array}{l} f(b)=f(a)+\frac{1}{2}(b-a)\left(f^{\prime}(a)+f^{\...
Vibrato's user avatar
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0 answers
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The sign of a determinant?

Let $$A = [a_{ij}] = \begin{bmatrix} 1& 1/2! & 1/3!\\ 1/2! & 1/3!& 1/4!\\ 1/3!& 1/4! & 1/5!\end{bmatrix}.$$ Let $A^{\circ r}=[a_{ij}^r]$ for $r>0$. We need to prove that $\...
VSP's user avatar
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0 answers
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Calculating maximum of the function using Mean Value Theorem

Let $A = [0,1]^2$ and $f(x,y) = (e^x cos y, e^x sin y)$. Using the Mean Value Theorem, determine the maximum value of $||f(x,y)||$ where $(x,y) \in A$. Explain why the maximum is attained on A. I have ...
mia's user avatar
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