Given smooth real-valued function $f: \mathbb{R}\rightarrow\mathbb{R}$ with $f(0) = 0$ and $f(1) = 1$ prove that there exist $x_1, x_2 \in [0, 1], x_1 \ne x_2$ such that  $\frac{1}{f'(x_1)} + \frac{1}{f'(x_2)} = 2$.   The only thing I could think of is noticing that by Lagrange's Mean Value theorem there exists $x'$ with $f'(x') = 1$: $$f(1) - f(0) = f(x')(1 - 0)$$  I can neither prove there exists another point with derivative being equal to $1$ not come up with another solution idea. Could you help me please?
Suppose that $f$ is continous on $[0,1]$ , differentiable on $(0,1)$ , and $f(0)=0$ and  $f(1)=1$.For every integer $n$ show  that there must exist $n$ distinct points $\alpha_1,\alpha_2\cdots,\alpha_n$ in that interval so that    $\sum \limits_{k=1}^n \frac{1}{f'(\alpha_k)} = n $   Is it theorem or i can show it with basic differentiation knowledge .Please give me hint .
Can a ring of positive characteristic have infinite number of elements? Comma usage with compound sentences {actually compound predicates as per Rob's answer} Rule 101 of compound sentences is that a coordinate conjunction must be accompanied by a comma most of the time when joining independent clauses, yet I stumble across sentences all that time that seem to abuse this rule. Normally, it's a case of the subject not being present in the second clause, as in the following example:      "They usually start out as an abbreviation, but turn into acronyms   when they become well known."   (taken from the Oxford Style Manual)   To me, this comma cannot be correct because the clause after the comma doesn't contain a subject and therefore cannot be independent (right?). As far as I'm concerned, it's not the 'Comma to set off a contrast rule because I can find plenty of other where 'and' is used and the subject's missing in the second clause.  what's the deal here?   Thanks in Advance for any responses I may receive! When we find the electric field between the plates of a parallel plate capacitor we assume that the electric field from both plates is $${\bf E}=\frac{\sigma}{2\epsilon_0}\hat{n.}$$ The factor of two in the denominator comes from the fact that there is a surface charge density on both sides of the (very thin) plates. This result can be obtained easily for each plate. Therefore when we put them together the net field between the plates is $${\bf E}=\frac{\sigma}{\epsilon_0}\hat{n}$$ and zero everywhere else. Here, $\sigma$ is the surface charge density on a single side of the plate, or $Q/2A$, since half the charge will be on each side.   But in a real capacitor the plates are conducting, and the surface charge density will change on each plate when the other plate is brought closer to it. That is, in the limit that the two plates get brought closer together, all of the charge of each plate must be on a single side. If we let $d$ denote the distance between the plates, then we must have $$\lim_{d \rightarrow 0}{\bf E}=\frac{2\sigma}{\epsilon_0}\hat{n}$$ which disagrees with the above equation. Where is the mistake in this reasoning?  Or more likely, do our textbook authors commonly assume that we are in this limit, and that this is why the conductor behaves like a perfectly thin charged sheet? Using Model to equally distribute points, spatially within known geometry using ArcGIS ModelBuilder? Is it possible to create a model in ArcGIS ModelBuilder where you simply state number of points and an area in the form of a polygon to get the model to try and equally space out the points based on the geometry of the polygon and number of points? What is the difference between object keys with quotes and without quotes? Could we provide some automatic feedback on (enough) unfriendly/rude comments? Finding the uniquely determined region of a PDE (a) Solve the equation $yu_x+xu_y=0$  with the condition $u(0,y) = e^{-y^2}$. (b) In which region of the xy plane is the solution uniquely determined?     I did the first part but I don't understand how to do part b.  I read   But I don't understand how the argument is based. A more clear explanation is appreciated Get specific GPS location somehow on windows 8 redraw GUI on didreceivelocalnotification when come from lock screen iphone Thick annulus (ring) in 3D Detecting pattern at the end of a line with grep Best way to return multiple values from a function? KeyboardViewer app is saved from this AppleScript.  on run {input, parameters}     if application "KeyboardViewer" is running then         quit application "KeyboardViewer"     else          activate application "KeyboardViewer"     end if     return input end run   Dock toggle save. Adding shortcut to above, needs some work...   Syntax error Can’t set application "KeyboardViewer" to some object. Access not allowed.  tell application "System Events"     set application "KeyboardViewer" to (keystroke "K" using control down) end tell When does a Möbius transformation map $\Im(z)>0$ to itself? Show Möbius transformation which maps $\Im(z)&gt;0$ to itself iff  $$ f(z)= \frac{az+b}{cz+d}\,,\,\,ad-bc&gt;0$$ and $a,b,c,d$ are real. Is it possible to write a template that changes behavior depending on if a certain member function is defined on a class?  Here's a simple example of what I would want to write:  template&lt;class T&gt; std::string optionalToString(T* obj) {     if (FUNCTION_EXISTS(T-&gt;toString))         return obj-&gt;toString();     else         return "toString not defined"; }   So, if class T has toString() defined, then it uses it; otherwise, it doesn't. The magical part that I don't know how to do is the "FUNCTION_EXISTS" part. What characters are allowed for the display name? I tried editing my display name and for some names it gives me this error:     Display Name can only contain a-z, 0-9, spaces, apostrophes or hyphens and must start with a letter or number   That is clearly not the case, because it will take the following names:   NullUserException -龍- NullUserException Ω    NullUserException ఠ_ఠ NullUserException ಠ_ಠ NullUserException இ_இ   But it won't let me do:   NullUserException  (╯°o°)╯ NullUserException ٩๏̯͡๏)۶ NullUserException ☜(ﾟヮﾟ☜) NullUserException อ้_อ้   So what's the regex used (assuming that's what's being used), so I can try out allowed names without actually changing it? Can anyone can suggest where I can freely download ArcObjects SDKs for ArcGIS?  I am using version 10.2. Finite groups with exactly one maximal subgroup I was recently reading a proof in which the following property is used (and left as an exercise that I could not prove so far). Here is exactly how it is stated.     Let $G$ be a finite group. Suppose it has exactly one maximal subgroup. Then $G$ is cyclic.   Now, does this mean exactly one conjugacy class of maximal subgroups? If it really means exactly one maximal subgroup, then what can we say about a finite group that possesses exactly one class of (at least two) maximal subgroups that are all conjugate to each other? Why Do Glueballs Have Mass, When Individual Gluons Are Massless? In $\mathbb{R}$ with standard topology the closure of the above set $M$ is $M \cup 0$. I fail showing this by purely topological means. I.e. not using the fact that this holds in the reals as a metric space and deduce it for the induced topology.  I assumed that there exist a closed subset $A$ containing $M$ but not 0. Hence, the complement of $A$ is an open subset containing 0. Does this already yield a contradiction or do I need a different approach?  Could you help me on this?