Questions tagged [mean-value-theorem]
Covers various forms of the Mean Value Theorem in calculus, including Lagrange's, Cauchy's, and Rolle's theorems, along with their applications in analysis and problem-solving.
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Generalizing a root existence problem for periodic functions
I recently came across the following problem:
Let $f: \mathbb{R} \to \mathbb{R}$ be a continuously differentiable periodic function with period $T > 0$ such that $\int_{0}^{T} f(x) \, \mathrm{d}x =...
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prove that there is no differentiable function $f:\mathbb{R}\to\mathbb{R}$ such that $f(f'(x))=x$, $\forall x\in\mathbb{R}$
Prove that if $f:\mathbb{R}\to\mathbb{R}$ is differentiable such that $f'$ is injective, then $f$ takes each value at most twice.
Using this (or otherwise) prove that there is no differentiable ...
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Solving Duhamel integral using the mean value theorem
In the system theory Duhamel integral is used to determine the response $y\left( t \right)$ of linear time-invariants systems to any input signal $x\left( t \right)$ based-on the system unit impulse ...
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Approximating error using the mean value theorem [duplicate]
I have been asked in my coursework to use the mean value theorem to approximate errors in calculation, such as calculating square roots of non-integers. I understand how to do the calculation without ...
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Mean Value Inequality for arbitrary norms on $\mathbb R^n$
Let $E$ be a normed real vector space and let $f\colon [a, b] \to E$ be continuous on $[a, b]$ and differentiable on $(a, b)$. If $E$ is an inner product space, or if $E = \mathbb{R}^n$ equipped with ...
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On proving $1-\sqrt{1-x^2}<\sqrt{1-x^2}-\sqrt{1-2x^2}$ when $0<x \le \frac{1}{\sqrt{2}}$by using the mean-value theorem
We want to prove the following using the concept of concave functions:
$$1-\sqrt{1-x^2}<\sqrt{1-x^2}-\sqrt{1-2x^2}, \quad \left(0<x \le \frac{1}{\sqrt{2}}\right).$$
The function $f(u)=\sqrt{1-u^...
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$\forall x \in \mathbb{R}, f'(x) > f(f(x))$, Prove that for all $x \geqslant 0$, $f(f(f(x))) \leqslant 0$.
I met this question when I did my analysis homework: Let $f$ be a continuously differentiable function on $\mathbb{R}$, and suppose that for all $x\in\mathbb{R}$, $f'(x) > f(f(x))$. Prove that for ...
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Must $g(4)$ satisfy any inequality if only $g(1)$, $g'(1)$, and $g''(1)$ are known?
Consider an everywhere differentiable function $g(x)$. Suppose we are given only the following information at the single point $x=1$:
$g(1) = 7$, $g'(1) = -3$, $g''(1)>0.5$
We are asked which ...
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Show that if $\frac{f'}{g'}$ increases, then $\frac{f}{g}$ is increasing
Let $f, g \in C^1(0, a]$, $f(0) = g(0) = 0$, $g(x) > 0$ and $g'(x) > 0$ $\forall x \in (0, a]$.
Show that if $\frac{f'}{g'}$ increases on $(0, a]$, then $\frac{f}{g}$ increases on this interval.
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A Problem on the Mean Value Theorem for Improper Integrals
Assume that $ \int_{1}^{+\infty} f(x) dx $ converges (as improper Riemann integral). Does there necessarily exist some $ \xi \in (1, +\infty) $ such that
$$
\int_{1}^{+\infty} \frac{f(x)}{x} dx = \...
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Does the Mean Value Theorem work for differentiable functions $f:E\to \mathbb{R}$ defined on any normed space $E$?
It is know that if $f:\mathbb{R}^n\to \mathbb{R}$ is differentiable and $[a,b]$ is the segment from $a$ to $b$, there is at least one number $c\in [a,b]$ such that $df(c)(b-a)=f(b)-f(a)$. The proof ...
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Some confusion in the range of mean value point of Second mean value theorem for definite integrals.
I am currently trying to solve a problem using the second mean value theorem for integrals. The process requires a strict condition on the range of the intermediate point—specifically, I want it to be ...
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let $f \in C^4([0,4])$ and $\int_0^4f(x)dx+3f(2)=8\int_1^3f(x)dx$, then prove existence of $c$ so $f^{(4)}(c)=0$
Let $f$ be a four times differentiable function on $\mathbb R$ and $f^{(4)}$ be continuous on $[0,4]$. Assuming that
$$\int_0^4f(x)dx+3f(2)=8\int_1^3f(x)dx,$$
prove that there exists $c\in(0,4)$ so ...
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Reconstructing a problem with typo
Previously I've posted this question here but was pointed out that the claim to be proven is false. I've brought this up to the lecturer who gave out this problem and was provided with a "...
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Existence of $c_1,c_2$ such that $\frac{f(a)f'(c_2)+f(c_2)f'(c_1)}{b-c_2}=f'(c_1)f'(c_2)$
Let $f:[a,b]\longrightarrow\mathbb R$ be a continuous and differentiable function on $[a,b]$, with $0\leq a<b$. Prove that there exist $c_1,c_2\in(a,b)$ such that
$$\frac{f(a)f'(c_2)+f(c_2)f'(c_1)}{...
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