# Tagged Questions

0answers
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### Din derivatives and fundamental theorem of calculus

I have been looking for some references concerning the fundamental theorem of calculus and Dini derivatives and I did not find it. I would like to know if given a locally Lipschitz function ...
0answers
41 views

### Second derivative wrt complex parameter

I'm facing an estimation problem and I need to calculate the Cramer-Rao Lower Bound of an estimator. So I have 2 unknown parameters: the amplitude of the signal $A$ and its direction of arrival $u$. ...
1answer
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### Find all solutions to a particular differential equation

Find all solutions on ${R}$ of the differential equation $y' = 3|y|^ \frac{2}{3}$ I believe I need to use separation of variables, but it will only work if the initial condition is nonzero. ...
3answers
48 views

### Does continuity imply existence of one sided derivatives?

From what I understand a derivative may not exist at a given point if the function is not continuous or the right and left side derivatives are not equal. Does that imply that if a function is ...
1answer
42 views

### Diferentiable function with non-differentiable inverse

Is it possible to define bijective function $f: \mathbb{R} \to \mathbb{R}$ that is differentiable in a point $x_0$ such that $f'(x_0) \ne 0$, but $f^{-1}$ is not differentiable in $f(x_0)$? I think ...
2answers
41 views

### Let $f(x)=\int_0^1|t-x|t~dt$ for all real $x$. Sketch the graph of $f(x)$, what is the minimum value of $f(x)$

Let $f(x)=\int_0^1|t-x|t~dt$ for all real $x$. Sketch the graph of $f(x)$, what is the minimum value of $f(x)$ I could not in any way understand how to approach this problem. I think I will be able ...
1answer
28 views

### Intuition and counterexamples for higher-order derivative test

In the higher-order test we keep differentiating a function till we find the n'th derivative (n being even) to be greater than or less than zero thereby identifying it as a minimum or maximum. My two ...
1answer
43 views

### Term wise differention

Consider $S(x) = \displaystyle{\sum_{k=1}^{\infty}} x^k k^2$. (a) Find an explicit formula for $S(x)$ on the interval $-1<x<1$ by repeated termwise differentiation of a geometric series. Be ...
1answer
69 views

### Are graphs of derivatives connected?

One of the basis results in real analysis is the Darboux's theorem, which says the derivative of a differentiable function has the intermediate value property. I've always been a bit dissatistied ...
2answers
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### Proof using Fermat's Theorem on stationary points?

It seems intuitive that, if a function differentiable on [a,b] is such that f'(a) < 0 < f'(b) then there exists some c in the open interval (a,b) such that f'(c)=0, but I can't prove it ...
0answers
17 views

### Characteristics of functions whose second derivative exists on an interval.

If a function $f:\left[a,b\right]\mapsto\mathbb R$ is differentiable on [a,b] and $f':\left[a,b\right]\mapsto\mathbb R$ is differentiable on [a,b] Then do the following statements have to be true? ...
1answer
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### Determine the values of $r$ where $f´(0)$ exist.

If $r>0$ is rational, let $f:\mathbb R\to \mathbb R$, whit $f(x)=x^r\cdot\sin(1/x)$ for all $x \neq 0$ and $f(0):=0$, determine the values of $r$ where $f´(0)$ exist. I dont know hoy can I ...
2answers
41 views

### Find where the function $k(x):=|\sin(x)|$ is differentiable and calculate its derivative

Find where the function$$k(x):=|\sin(x)|$$ is differentiable and calculate its derivative. I have started, by trying to make a function by parts, because of the absolute value, getting this: ...
1answer
17 views

### Extending a polynomial function on an interval to be infinitely differentiable on all of R

If $f:(a,b) \to \mathbb{R}$ is a polynomial function, can it be extended to $g:\mathbb{R} \to \mathbb{R}$ such that g is infinitely many times differentiable and it is NOT the same polynomial? What ...
0answers
33 views

### Differentiablilty of composition functions

Two questions I suppose. One comes from a test I recently took that I didn't quite get/understand the method I should be using (or even how I should proceed) Let $f:R^2 \rightarrow R$ s.t $f$ is an ...
2answers
41 views

### Differentiable function in n dimensions

If a function $f:\mathbb R^{n} \rightarrow \mathbb R^m$ is differentiable at a point $a$ can we say that there is a neighbourhood of $a$ such that $f$ is locally Lipschitz? (i.e. there is some ...
1answer
40 views

### Partial derivative on convex set

If we have a function $f:U \rightarrow R$ ($U \subset R^n$) which is partially differentiable on a convex set U with $\frac{\delta f}{\delta x_1} = 0$ for all $x \in U$. How can we prove that $f$ ...
2answers
46 views

### Increasing function on small interval given positive derivative

Suppose that $f'(0)>0$. Does it imply that there exists a $\delta > 0$ such that $f$ is increasing on $[0,\delta]$? I think this is false and I've been trying to think of a counter example. I ...
2answers
66 views

### If $f$ is differentiable and $\lim_{x→0} f'(x) = L$, then $f'(0) = L$.

True/False. (c) If $f$ is differentiable on an interval containing zero and if $\lim_{x→0} f'(x) = L$, then $f'(0) = L$. 1. How to presage proof by contradiction? Proof by contradiction. ...
2answers
78 views

### Show that $f$ is not increasing on any interval containing $0$

$f:R\to R$, $f(x)=x^2\sin(1/x)+x$ if $x\ne 0$ and $0$ if $x=0$ In the first part of this problem, I showed that $f'(0)>0$ The second part of the problem is this: Show that $f$ is not increasing ...
2answers
139 views

3answers
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### Function which is never its own ($n^{th}$) derivative?

Is there any real-valued function $f(x)$ of a real variable $x$ with $n^{th}$ and $m^{th}$ derivatives never equal for nonequal nonnegative $m$ and $n$ and where the $n^{th}$ derivative of $f$ never ...
1answer
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1answer
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### Differentiability in $\mathbb R^n$

Let $U\in \mathbb{R}^n$ be open, and let $f:U\to \mathbb{R}^m$, and let $a\in U$. Let $\|\cdot\|'$ be a norm on $\mathbb{R}^n$, and let $\|\cdot\|''$ be a norm on $\mathbb{R}^m$. Prove that $f$ is ...
7answers
8k views

### What function can be differentiated twice, but not 3 times?

In complex analysis class professor said that in complex analysis if a function is differentiable once, it can be differentiated infinite number of times. In real analysis there are cases where a ...
1answer
42 views

### Tangent vector to a curve on a manifold

If one has a curve $\sigma : (-1,1) \rightarrow M$, where $M$ is a smooth manifold, the tangent vector in $\sigma(0)$ is usually defined as $$\sigma'(0) (f) = \dfrac{d f \circ \sigma}{dt} \Big|_0,$$ ...
1answer
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1answer
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### Show a function is not bounded

Let $f(x):(0,\infty)\rightarrow \mathbb{R}$, $f$ is differentiable. It is given that: $\mathop {\lim }\limits_{x \to \infty } f'(x) = L > 0$. Show $f$ is not bounded. I've seen a proof ...