Prove or disprove.  If $f: \mathbb{R} \rightarrow \mathbb{R}$ is a differentiable function, $f'(x) &gt; f(x)$ for all $x$, and $f(0) = 0$, then $f(x) &gt; 0$ for all $x&gt;0$.  If I could be pointed in the right direction, it would be highly appreciated.
I was wondering if there are functions for which $$f'(x) &gt; f(x)$$ for all $x$. Only examples I could think of were $e^x - c$ and simply $- c$ in which $c &gt; 0$. Also, is there any significance in a function that is always less than its derivative?    Edit: Thank you very much for all the replies. It seems almost all functions that apply are exponential by nature...  Are there more examples like - 1/x?  Again are there any applications/physical manifestations of these functions? [for example an object with a velocity that is always greater than its position/acceleration  is always greater than its velocity]
I'm looking for a 90s (or early 2000s) scifi (mini)series where a boy and old man chase a black guy in metal suit. I think they were going through different dimensions. I remember part of it, but I don't think I ever saw how it ended. I remember my friend telling me there was a twist at the end. I think they were trying to find something and he was always ahead of them.. Order of execution of jquery document ready Generating random numbers in C Query a region within a larger region with GeoDjango REST API I have a working Django REST API which can query all the counties in a country. //My model class kenya_counties(models.Model):   objectid = models.BigIntegerField()   id_field = models.BigIntegerField()   county_nam = models.CharField(max_length=80)   const_code = models.BigIntegerField()   constituen = models.CharField(max_length=80)   county_cod = models.BigIntegerField()   shape_leng = models.FloatField()   shape_area = models.FloatField()   geom = models.MultiPolygonField(srid=4326)    def __str__(self):       return self.county_nam  //My Serializer class CountiesSerializer(serializers.ModelSerializer):   class Meta:      model = kenya_counties      fields = ('objectid', 'id_field', 'county_nam', 'const_code', 'constituen', 'county_nam', 'county_cod', 'shape_leng', 'shape_area')  //My view @api_view(['GET', 'POST']) def counties(request):    if request.method == 'GET':        data = kenya_counties.objects.all()         serializer = CountiesSerializer(data, context={'request': request}, many=True)         return Response(serializer.data)     elif request.method == 'POST':        serializer = CountiesSerializer(data=request.data)        if serializer.is_valid():            serializer.save()            return Response(status=status.HTTP_201_CREATED)                 return Response(serializer.errors, status=status.HTTP_400_BAD_REQUEST)  //My url     re_path(r'^api/gis/counties/$', views.counties),  I get . How can I query just one county and display it on a react-leaflet map as shown below? &lt;MapContainer center={position} zoom={13} scrollWheelZoom={false}&gt;     &lt;TileLayer       attribution='&amp;copy; &lt;a href=&quot;http://osm.org/copyright&quot;&gt;OpenStreetMap&lt;/a&gt; contributors'       url=&quot;https://{s}.tile.openstreetmap.org/{z}/{x}/{y}.png&quot;     /&gt; &lt;/MapContainer&gt; Invariance of a maximally entangled state under unitary operation $U \otimes U^\dagger$ Difference between $:=$ and $=$ I am sorry if this is quite elementary question. But I always think, that why we use $:=$ at some places, instead of $=$. Is there any fundamental difference between these two? Before reading Terry Tao's blog (4 months ago), I had never seen a symbol like this (:=). It seems like to deleted too much files by rm -r and now I have the error of "the system is running a low-graphic mode" in Ubuntu and I have to work in Windows because of that.  However I can load it in recovery mode. I need to copy the files from Downloads folder to a flash drive. Downloads folder has them as I can see by ls command.   When I plug in a flash drive, there will be some messages saying that everything is ok (as I understood) and it's plugged. But when I go to /media/my_name and do ls then I don't see anything. I don't see anything in /run/media/my_name either(I'm not sure if should though).  I can't connect to the Internet for some reason. Ubuntu was installed by wubi.exe file.  How do I save the content of my Downloads folder to a flash drive? Pgfplots with custom axis markers Assume you have a trigonometric function, say for simplicity sin(x). When plotting the function, how can one place the x-axis markers on 0, pi/2, pi, 3pi/2, 2pi, instead of the default numeric values that pgfplots uses? I've been trying a simple recipe I found on the internet, which is a simple recipe consisting mostly of gelatin.  The problem is that the end result is basically just a fruit-flavoured jelly (Jell-O?), rather than the more chewy gummy-bears (Haribo) style of jelly confectionery.  Can anyone suggest what I need to do to make them less springy and more chewy.  It's all difficult to articulate!  Here's what I used for the first test-batch:   12 tbsp sugar 12/3 cup fruit juice 8 tbsp Golden Syrup 8 tbsp gelatin   soften the gelatin in 1/2 cup of cold water. place sugar, syrup and the juice in a pan and gently heat until sugar dissolves. Stir in the gelatin and stir until dissolved. Pour into moulds and wait until set.    Thanks Derivative of $f(x)=|x|$ How can Innodb ibdata1 file grows by 5X even with innodb_file_per_table set? I have innodb_file_per_table set and just today my ibdata1 file jumped from 59M to 323M after I made several changes to an 800M table to reduce it to about 600M. That particular table's .ibd file was reduced but the server's ibdata1 file went crazy. Any ideas? Is it safe to check floating point values for equality to 0? I know you can't rely on equality between double or decimal type values normally, but I'm wondering if 0 is a special case.  While I can understand imprecisions between 0.00000000000001 and 0.00000000000002, 0 itself seems pretty hard to mess up since it's just nothing.  If you're imprecise on nothing, it's not nothing anymore.  But I don't know much about this topic so it's not for me to say.  double x = 0.0; return (x == 0.0) ? true : false;   Will that always return true? Is accessing a variable in C# an atomic operation? I've been raised to believe that if multiple threads can access a variable, then all reads from and writes to that variable must be protected by synchronization code, such as a "lock" statement, because the processor might switch to another thread halfway through a write.  However, I was looking through System.Web.Security.Membership using Reflector and found code like this:  public static class Membership {     private static bool s_Initialized = false;     private static object s_lock = new object();     private static MembershipProvider s_Provider;      public static MembershipProvider Provider     {         get         {             Initialize();             return s_Provider;         }     }      private static void Initialize()     {         if (s_Initialized)             return;          lock(s_lock)         {             if (s_Initialized)                 return;              // Perform initialization...             s_Initialized = true;         }     } }   Why is the s_Initialized field read outside of the lock? Couldn't another thread be trying to write to it at the same time? Are reads and writes of variables atomic? Prove $\gcd(a,b,c)=\gcd(\gcd(a,b),c)$ for $0\ne a,b,c\in \Bbb{Z}$. I tried solving it with sets but I sense there are some details I am missing. I would truly appreciate your reference. Upgrade from MaaS 2.6.2 to MaaS 2.7 Remove extra header lines from file, except for the first line I have a file that looks like this toy example.  My actual file has 4 million lines, about 10 of which I need to delete.  ID  Data1  Data2 1    100    100 2    100    200 3    200    100 ID  Data1  Data2 4    100    100 ID  Data1  Data2 5    200    200   I want to delete the lines that look like the header, except for the first line.  Final file:  ID  Data1  Data2 1    100    100 2    100    200 3    200    100 4    100    100 5    200    200   How can I do this? Can you please explain this to me?          Could you please explain this to me?   I am unable to figure out which to use which situation. I did google, and some posts say they are both the same, even if the second one is more formal. Other posts say could is the past tense of can, but in the above example, I don't think could is used as past tense of can.  What is correct way to use these words? Is it possible to get total render time via Python code when render’s done? How to upgrade from 18.10 to 19.10, using the command line? I'm trying to upgrade from 18.10 to 19.10, using sudo do-release-upgrade however I only get an error: An upgrade from 'cosmic' to 'eoan' is not supported with this tool. I performed the steps to update my sources.list to point to old-release server, as suggested here:  This did not help. I get the same error. How does "Reputation" work? On Stack Exchange, users may gain a certain level of reputation.  What does reputation do? How can a user gain or lose reputation?   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