Let $f: \mathbb{R} \to \mathbb{R}$ be differentiable everywhere, $f’(x) = f(x)$ for all $x$ and $f(0) = 1$ then show that $f(x)$ cannot be zero for any $x$. I can see that this will not be the case for at least $x$ strictly greater than zero. I can also see that if $f$ is zero for some $x &lt;0$ then we can’t have $f(0) = 1$ but I don’t know how to formally prove the proposition. Please help me out.
The statement is: $f$ is a real fucntion on $\mathbb R$. Then if $f'(x)=f(x)$ and $f(0)=1$, then $f(x)\neq 0.$
copy and paste doesn't work in terminal Derive equation for a cantilever in SHM I am having difficulty with this proof. I have previously proved that for an automorphism on the extension field $K$ of $F$ that when restricted to $F$ is the identity automorphism must take roots of $f(x)\in F[x]$ in $E$ to roots of $f(x)$ in $E$. I was thinking of using the minimal polynomial $x^{3}-2\in\mathbb{Q}[x]$ and the result of previous problem I solved, but I suspect there are automorphisms that don't fix the subfield, so the result can't be used or I need to prove that any automorphism of an extension field must fix the subfield that is being extended or always sends roots in $E$ to roots in $E$.  I am not asking for a solution to this, but maybe a hint or pointer in the right direction. This problem is introduced in the text prior to Galois theory, so please keep that in consideration. I am using two books for my calculus refresher.   Thomas' Calculus  Higher Math for Beginners by Ya. B. Zeldovich   My question is : When applying Integral Calculus for calculation of volumes of solids, generated by curves revolved around an axis, we use slices of 'cylinders' to approximate the volume of the resulting solid and then integrate the sum of those infinitesimal cylinders. However, when we are using the same techniques to calculate the surface area of the surfaces generated by revolving curves around an axis, we consider the 'slope' of the differential length of curve 'dx', calculate the total length of the curve and derive the required surface area.  Are we not supposed to use the same 'slope' for calculating the volumes of the infinitesimal 'cylinders' for calculation of volumes? Shouldn't we use 'sliced portions of 'cones' as the infinitesimal volumes?? When it come to calculation of volumes of solids of revolution, why are we neglecting the slope of the curve for the differential length and simply assume that it is an infinitesimal cylinder??  Ex: Let us say we want to calculate the surface area and the volume of the solid generated when the parabola y = 10 . x^2 is revolved about the y-axis, with limits of x from 0 to 5.  In such cases, when calculating the volume of the solid, we consider infinitesimal 'cylinders', ignoring the 'slope' of the curve for the differential element 'dx', but when calculating the surface area, we consider the 'slope' of the differential element 'dx'. How can I best describe "making a new discovery"? Can a shell script set environment variables of the calling shell? As I understand this, firewalls (assuming default settings) deny all incoming traffic that has no prior corresponding outgoing traffic.  Based on  and , reverse SSH tunneling can be used to get around pesky firewall restrictions.  I would like to execute shell commands on a remote machine. The remote machine has its own firewall and is behind an additional firewall (router). It has an IP address like 192.168.1.126 (or something similar). I am not behind a firewall and I know the remote machine's IP address as seen from the Internet (not the 192.168.1.126 address). Additionally, I can ask someone to execute ssh (something) as root on the remote machine first.  Could anyone explain me, step by step, how reverse SSH tunneling works to get around the firewalls (local and remote machines' firewalls and the additional firewall between them)?  What is the role of the switches (-R, -f, -L, -N)? Error in Obj-C : Expected identifier or '(' Determine if $.ajax error is a timeout What causes yogurt in sauces to split? How to prevent it? A lot of my favorite curry recipes have a yogurt based sauce in them, but on a pretty regular basis when I make them, the yogurt ends up splitting into basically curds and whey. What causes that, and how can I prevent it?  As an illustrative example, last night I made a simple tofu curry as follows:   warm evoo in a pan add chopped onion, saute briefly add curry seasonings, allow to warm/season the oil add cubed tofu, toss to coat cook for a while, tossing periodically to lightly brown the cubes turn heat off add plain greek yogurt stir to combine   In the few minutes it took to finish the rest of the meal and start plating, the yogurt had separated so I had a clumpy, lumpy, yogurt soup instead of a smooth creamy sauce. I'm fitting a multiple linear regression model between 4 categorical variables (with 4 levels each) and a numerical output. My dataset has 43 observations.  Regression gives me the following $p$-values from the $t$-test for every slope coefficient: $.15, .67, .27, .02$. Thus, the coefficient for the 4th predictor is significant at $\alpha = .05$ confidence level.  On the other hand, the regression gives me a $p$-value from an overall $F$-test of the null hypothesis that all my slope coefficients are equal to zero. For my dataset, this $p$-value is $.11$.  My question: how should I interpret these results? Which $p$-value should I use and why? Is the coefficient for the 4th variable significantly different from $0$ at the $\alpha = .05$ confidence level?  I've seen a related question, , but there was an opposite situation: high $t$-test $p$-values and low $F$-test $p$-value. Honestly, I don't quite understand why we would need an $F$-test in addition to a $t$-test to see if linear regression coefficients are significantly different from zero. Remove rogue app from menu bar htaccess subdomain Show $\lim_{h\to 0} \frac{(a^h-1)}{h}$ exists without l'Hôpital or even referencing $e$ or natural log Unable to modify files/folders in NTFS partition How do you add tags to blocks so they can only be placed on 1 type of block? How do you deserialize json properties that are reserved words in apex? Possessive form for a surname ending with "z" What is the proper possessive form for a surname that ends with “z”? Is it z’ or z’s? How do I join a Windows Active Directory? We have a "mix-mode" Lab. and we want to migrate to Ubuntu, from Windows.  In this process, how do we add Ubuntu workstations to a Windows Domain / Active Directory ('AD') setup? Assume you have a flat table that stores an ordered tree hierarchy:  Id   Name         ParentId   Order  1   'Node 1'            0      10  2   'Node 1.1'          1      10  3   'Node 2'            0      20  4   'Node 1.1.1'        2      10  5   'Node 2.1'          3      10  6   'Node 1.2'          1      20   Here's a diagram, where we have [id] Name.  Root node 0 is fictional.                          [0] ROOT                           /    \                [1] Node 1          [3] Node 2               /       \                   \     [2] Node 1.1     [6] Node 1.2      [5] Node 2.1           /            [4] Node 1.1.1   What minimalistic approach would you use to output that to HTML (or text, for that matter) as a correctly ordered, correctly indented tree?   Assume further you only have basic data structures (arrays and hashmaps), no fancy objects with parent/children references, no ORM, no framework, just your two hands. The table is represented as a result set, which can be accessed randomly.   Pseudo code or plain English is okay, this is purely a conceptional question.  Bonus question: Is there a fundamentally better way to store a tree structure like this in a RDBMS?    EDITS AND ADDITIONS  To answer one commenter's ('s) question: A root node is not necessary, because it is never going to be displayed anyway. ParentId = 0 is the convention to express "these are top level". The Order column defines how nodes with the same parent are going to be sorted.  The "result set" I spoke of can be pictured as an array of hashmaps (to stay in that terminology). For my example was meant to be already there. Some answers go the extra mile and construct it first, but thats okay.  The tree can be arbitrarily deep. Each node can have N children. I did not exactly have a "millions of entries" tree in mind, though.  Don't mistake my choice of node naming ('Node 1.1.1') for something to rely on. The nodes could equally well be called 'Frank' or 'Bob', no naming structure is implied, this was merely to make it readable.  I have posted my own solution so you guys can pull it to pieces.