Questions tagged [mean-value-theorem]
The mean-value-theorem tag has no usage guidance.
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Gradient and intermediate value theorem
I'm having trouble with an optimization problem. I have a continuous and concave objective function $\mathcal{O} : \mathbb{R}^n \longrightarrow \mathbb{R}$. I have strong evidence (numerical ...
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Arc length derivation [closed]
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Why does the hypothetical X value Xi that would equal the avg slope of a section get replaced by X? I don't remember using the mean value theorem in the definition of the ...
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$ {f}^{\prime }\left( \xi \right) = {2024}\left( {f\left( \xi \right) - f\left( 0\right) }\right) $
Let $f\left( x\right)$ be differentiable on $\lbrack 0, 2\rbrack$ , ${f}^{\prime }\left( 1\right) = 0$, Prove that there exists $\xi\in\left( {0, 2}\right)$ such that
\begin{aligned} {f}^{\prime }\...
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Problem 237 "Mathematical Quickies:270 Stimulating Problems with Solutions" Particle Movement
I was solving "Mathematical Quickies:270 Stimulating Problems with Solutions" when I came across a very peculiar question (Problem 237):
A particle moves in a straight line starting from ...
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How to prove that the area of a continuous function $ f\left( x \right) $ enclosing a curved trapezium with the x-axis can be n-equalised
First is a simple problem for a continuous function $f(x)$ defined on the interval $[0,1]$ with $f(x)>0$. Prove that: on the interval $[0,1]$, there exists $x=x_0$ such that $x=x_0$ bisects the ...
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Stein Complex Analysis Proof of Chapter 6 Proposition 2.5
In Stein's Complex Analysis book, within the proof of Chapter 6 Proposition 2.5, the following claim is made:
For $s = \sigma + it \in \mathbb{C}, n \geq 1$, then
$$\left| \frac{1}{n^s} - \frac{1}{x^s}...
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Let $f(x):\mathbb{R}\to [-1,1]$ be twice differentiable and $f(0)^2+f'(0)^2=4$, then p.t. $\exists x_0$ s.t. $f(x_0)+f''(x_0)=0$ but $f'(x_0)\ne 0$ [duplicate]
The actual question is a multiple correct MCQ, but this was the only part I was having trouble with. I also can't fully attest for the correctness of the question, although my answer key does put this ...
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Show that a sequence generated by mean value theorem is approaching to $a$
I was trying to give a new proof Taylor’s series by taking $f(x)-\sum^{n}_{i=0}\frac{f^{(n)}(a)}{n!}(x-a)^{n}$ and applying mean value theorem n times. To make this complete, this problem is needed to ...
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Why is it that $f(x)>g(x)$ does not imply $f'(x)>g'(x)$ but it implies that $\int f(x) > \int g(x)$?
This is a question, I was solving
Background:
Let $$f(x)=\int^x_0\frac{t^2}{1+t^4}dt-2x+1$$
Then $$f'(x)=\frac{x^2}{x^4+1}-2=\frac{x^2-2x^4-2}{1+x^4}$$
Which is always negative$\implies$function a ...
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If derivative of $f$ has continous extension, $f$ is continously differentiable
Let $U$ be an open subset of $\mathbb{R}$, and let $x_0 \in U$.
Let $f$ be a continous function from $U$ to $\mathbb{R}$, which is continously differentiable everywhere on $U$, except for $x_0$.
Show ...
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Question regarding integral mean value theorem
I am sorry for cross-posting a question that I asked on mathoverflow. It is about the integral form of the mean value theorem which states that for the range $[a,b]$ for two continuous functions $f$ ...
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Is there a short proof of the Second Mean Value Theorm for Integrals (strong, preferably asymmetric version)
The parenthesis in the title comes from the fact that there are essentially six versions of the conclusion in what may be called 2nd Mean Value Th. for Int. - not including special variants like those ...
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Is my geometrical "proof" of CMVT correct?
In his A Course of Pure Mathematics, GH Hardy proves and gives a "geometrical demonstration" of the LMVT, which goes nearly like this:
THEOREM. If $\phi(x)$ is continuous on $[a,b]$ and ...
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Possible values of the integral of the function with the information of derivative
Let $f: [0,1] \to \mathbb{R}$ denote some continuously differentiable function such that $f '(x)$ is continuous and $ \int_0^1 f(x)dx=0$. If $\max_{x\in[0,1]}|f'(x)|=24$, then what are some/all of the ...
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early calculus question finding the root of $\cos(x) +2x -5$
Given $f(x)= \cos(x)+2x-5$ find c such that $f(c)=0$. The x-intercept.
The problem is from a calculus class.
I got as far as noting that when $f(x)=0$, $\cos(x) = 2x-5$. Therefore,
$$
-1 \le 2x-5 \...
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