# Questions tagged [mean-value-theorem]

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• 397
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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 ...
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1 vote
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• 143
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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 ...
• 521
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### 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 ...
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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 ...
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1 vote
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### 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 ...
• 5,666
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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)...
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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 $f(c) = \frac{1}{b-a} \int_a^b f(x) dx$...
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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, $f(U)$ ...
• 397
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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 $b$. (...
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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 $\ell$ ...
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### 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 ...
• 127
1 vote
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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 ...
• 49
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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 ...
124 views

### 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 ...
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### 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 ...
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1 vote
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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-...
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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 ...
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### 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 ...
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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: $$f'(c) = \frac{|f(b)-f(a)|}{|b-a|} \ge 0.$$...
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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 $c$, $\frac{d}{dx}g(x) > 0$.
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### 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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### 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 $|f(x)-f(y)|\le|x-y|$ ...
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1 vote
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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 ...
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1 vote
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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 ...
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1 vote
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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 "...
• 401
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