# Tagged Questions

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### Where is the error in my proof that all derivatives are continuous?

I know that this can not be true due to counter-examples but I don't know where the error in my reasoning is. Assumption: If $f(x)$ is differentiable in $\mathbb{R}$ then the derivative $f'(x)$ is ...
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### Question about Peano form of the remainder

Let $f(x)$ be a real-valued function defined on a closed interval [a, b], differentiable on the open interval (a, b) $n-1$ times. $x_0$ belongs to [a, b]. Suppose that we ...
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### A strange (!) behaviour of differentiability

I see by drawing diagram that $y=\max (0,\sin x)$ is not differentiable at some points. But $y=(\max (0,\sin x)) ^ 2$ is . How to explain/prove it ? Am I missing something easy ? If $f$ is not ...
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### Evaluate position of first secondary maximum of $\frac{\sin N (x/2)}{\sin (x/2)}$

The function $$f(x) = \displaystyle \left | \frac{\sin \left( N \displaystyle \frac{x}{2} \right)}{\sin \left( \displaystyle \frac{x}{2} \right)} \right |$$ when evaluated for $x > 0$, has its ...
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### Characterization of differentiable functions from $\mathbb{R}^m$ to $\mathbb{R}^n$.

Let $U\subset\mathbb{R}^m$ be an open set. Consider a function $f:U\to\mathbb{R}^n$ and a point $a\in U$. I need help to prove that the following sentences are equivalents. (a) There exists a ...
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### relations between differential, partial derivative, directional derivative

I am a bit lost. Could you explain me relations between differential, partial derivative, directional derivative? I mean that I need some theorem and proofs that for example if differential exists ...
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### Newton derivative of the distance function in $\Bbb R^2$

If we consider the distance function $d$, where $d(x)=dist(x,\partial\Omega)=\inf_{y\in\Omega}\|x-y\|_2$, how would one calculate the derivative in some direction $v$, i.e. ...
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### Extreme value problem, maximize ratio of volume to surface area

For a cylindrical can, how to choose the ratio of the height to the radius such that the ratio of the volume to the surface area gets maximized? The volume ist $V = \pi r^2 h$ and the surface ...
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### $F(x) = \int_a^b \frac{x^2}{1+2\sin^3(t) + \sin^6(t) } dt$

Let $F(x) = \int_a^b \frac{x^2}{1+2\sin^3(t) + \sin^6(t) } dt$ i have to calculate the derivative of $F(x)$ with respect to $x$. Let $g(x) = \frac{x^2}{1+2\sin^3(x) + \sin^6(x)}$ then $g$ is ...
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### Will antiderivative always be differentiable?

Suppose f(x) is continuous on [0,1]. Obviously, such a function will be integrable. Will antiderivative be always differentiable on (0,1)? The answer is "Yes" by the Fundamental Theorem of Calculus. ...
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### $f$ a differentiable fucntion in $[a,b]$ with $f´(a) < C < f´(b)$

Let $f$ a differentiable fucntion in $[a,b]$, suppose the existence of a point $C$ with $f´(a) < C < f´(b)$ how can i deduce that given the function $g(x) = f(x) - C(x-a)$ then exist a pint ...
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### Prove an inequality (Using Taylor expansion)

Prove: $\frac{x}{1+x} < \ln(1+x) < x$. I thought a good practice would be to prove it using Taylor Expansion. Here's my try: $$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3}...$$ The n=1 ...
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### Local minimum implies local convexity?

Consider a real function $f$, and suppose it has a local minimum at $a\in \mathbb R$. It typically looks like What hypotheses can be added to $f$ so that there is some $\epsilon >0$ such ...
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### if $f$ is differentiable at $x_0$ then the limit exists

Let $f$ differentiable at $x_0$. Show that the following limit exists $$\lim_{h\rightarrow0} \frac{f(x_0+h)-f(x_0-h)}{h}$$ If $f$ is differetiable at $x_0$ then it's one-sided derivative exists ...
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### $F(x) = \int_0^{x} t^2 e^{t^2}dt$

Let $y_0 = f''(2) + f'(1) + f(0)$ if $f$ is a real function defined by $f(x) = \int_0^{x} t^2 e^{t^2}dt$. How can I calculate the value of the expression $y_0$. I tried use the fundamental theorem ...
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### Prove there's $x_0$ such that $f'(x_0)=0$

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ differentiable at $\mathbb{R}$ and: $$\lim_{x\rightarrow \infty}\left( f(x)-f(-x) \right) = 0$$ Show there's $x_0$ such that $f'(x_0) = 0$. I tried to use ...
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### $F(x) = \int_0^2 \sin(x+l)^2\ dl$

Consider the function : $F(x) = \int_0^2 \sin(x+l)^2\ dl$, calculate $\frac{dF(x)}{dx}|_{x=0}$ the derivative of $F(x)$ with respect to $x$ in zero. Let $g(x) = \sin (x)$ and $h(x) = (x+l)^2$ then ...
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### simple way to show $|| \partial_x \int_{B(x,\epsilon)} \frac{x-y}{|x-y|^3} f(y) dy||_{\infty} = O(||f||_{\infty})$ in $\mathbb{R}^3$

We are set in $\mathbb{R}^3$. Let $f: \mathbb{R}^3 \rightarrow \mathbb{R}$ be a $C^1_0$ function, i.e. continuously differentiable with compact support. Let $\epsilon > 0$ be small. I need to show ...
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### Real Analysis Question about Taylor's Thm & High order derivatives

The question is: Let $I$ be a subset of $\mathbb{R}$ be an open interval and let $f\colon I\rightarrow\mathbb{R}$ be differentiable on $I$, and suppose $f''(a)$ exists at $a\in I$. Show that ...
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### Real Analysis. Derivative Question Help

The question is: Let $I$ be an interval. Prove that if $f$ is differentiable on $I$ and if the derivative $f'$ is bounded on $I$, then $f$ satisfies a Lipschitz condition on $I$. My attempt: As ...
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### Real Analysis Use of Darboux's Theorem Help

The question I'm trying to answer is: Let $I$ be an interval and let $f:I\rightarrow\mathbb{R}$ be differentiable on $I$. Show that if the derivative $f'$ is never 0 on $I$, then either $f'(x)>0$ ...
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Verify if the function $f(x,y)$ is differentiable in $(x,y) = (0,0)$ and $(x,y) \ne (0,0)$ $$f(x,y) = \begin{cases} 2xy (\frac{x^2 - y^2}{x^+ y^2}) & x^2 + y^2 \ne 0 \\[4pt] 0 & other ... 1answer 46 views ### Pointwise derivative Please I am struggling to find a proper definition of a pointwise derivative and also what is the difference between a pointwise derivative and the "classical" derivative. Can someone please point me ... 1answer 49 views ### Modifications of Weierstrass's continuous, nowhere differentiable functions Recalling how nowhere continuous functions such as the Dirichlet function can sometimes be modified on a \lambda-null set of points (in this instance, a countable set) to become everywhere ... 1answer 44 views ### Directional derivatives exist for function neither continuous nor differentiable at the point they exist It's an old past paper with no mark scheme, this always makes me a slightly afraid of exploring unaided. The function is: f(x,y)=\frac{xy^2}{x^2+y^6} if (x,y)\ne(0,0) else 0. I am to show all ... 1answer 51 views ### Question about Differentials I am reading the book "Advanced Calculus" written by Kaplan, and here is what I have: Suppose that y(x) is a differentiable function at x = x_0. Then, we can write y(x_0+\Delta x) = y(x_0) + ... 2answers 50 views ### Trying to solve a Taylor series problem I have a Taylor series problem, well more precisely a Maclaurin series. I am trying to find convergence of: f(x) = e^{x^3} + e^{{2x}^3} Okay here goes:$$f'(x) = 3xe^{x^3} + 6x e^{{2x}^3} ...
Having the following inequality, for a real-valued function $f$ which is twice differentiable: $f(a+h)-f(a)\geq f(a)-f(a-h)$ for any $a \in\mathbf{R}$, $h > 0$. and assuming that $f$ is bounded, ...