Problem: Prove that if $f$ is a twice differentiable function with $f(0)=0$, $f(1)=1$ and $f'(0)=f'(1)=0$, then $|f''(x)| \geq 4|$ for some $x \in [0, 1]$.  I tried to get a contradiction from $-4 &lt; f''(x) &lt; 4$ by integrating the inequality from $0$ to $1$, but got nothing useful. Please help.
Let $f(x)\in C^2:[0,1]\rightarrow\mathbb{R}$ satisfy $f(0)=0,f(1)=1,f'(0)=f'(1)=0$, prove that: $$\max_{x\in[0,1]}|f''(x)|\ge4$$  By using Taylor series I can prove that $\max_{x\in[0,1]}|f''(x)|\ge2$, but I cannot go further.
Some mesh-edges in object-mode aren't displayed So I have been extruding a cube to make a low poly banana and when I go from edit to object mode half the edges don't appear as well as half is dark.      I tried looking for an solution couldn't find one and its making me feel like I am doing something wrong so I decided to come here. Why am I getting strange results when using a power meter to measure CFLs? Is it possible to see a list of the settlers in a given settlement and the jobs they are assigned to? It would make it easier to pick a settler for a new job. How long can I store soaked beans before cooking? Why is lock(this) {...} bad? The  says that public class SomeObject {   public void SomeOperation()   {     lock(this)     {       //Access instance variables     }   } }  is &quot;a problem if the instance can be accessed publicly&quot;. I'm wondering why? Is it because the lock will be held longer than necessary? Or is there some more insidious reason? Why is Lamarck's evolutionary theory considered incorrect? Many sources postulate without much argumentation that Lamarck's evolutionary theory is "obviously incorrect". A classic example provided in such sources is the one with giraffes, in which Lamarck argued that the necks of these animals became long as a result of continually stretching to reach high foliage. This mechanism for adaptive change in response to the environment is considered wrong.  Now the counterargument to Lamarck's ideas comprises modern findings in genetics. So it is stated that organisms pass down traits through predetermined genetic information, not based on environmental adaptations during their lifetime. Furthermore, genetic research shows that living organisms cannot alter their genetic material as needed.   Doesn't the latter contradict the evolutionary theory itself, which states that nature (i.e. environment) shapes species through natural selection in the environmental milieu? If it is not the environmental/natural factor that is responsible for genetic evolution, then what is? If genetic material were to be unchangeable, then how would adaptation to the environment and consequent evolutionary change would ever be possible? The genotype (following this logic) would remain the same. And lastly, maybe giraffes do not develop long necks at such a rapid pace (as Lamarck probably thought) by simply stretching their necks in one generation; however, what, if not neck stretching, is responsible for their consistently longer necks throughout thousands or millions of years? Hadn't the environment gradually shaped their genotype? Similarly, if I run more often, maybe my children will not become more fit, but isn't it just a matter of time (thousands of years) that if all of us consistently run a lot, then our offspring will eventually become more fit? Wouldn't our voluntary efforts eventually enhance our genetics? I do not see any problem with Lamarck's logic at all, except probably the time frame. Calculating days of week given a week number Given a week number, e.g. date -u +%W, how do you calculate the days in that week starting from Monday?  Example rfc-3339 output for week 40:  2008-10-06 2008-10-07 2008-10-08 2008-10-09 2008-10-10 2008-10-11 2008-10-12 Why upload to academic preprint sites like arXiv? The recent question  made me realize that I don't have any understanding of why one would upload to a preprint site.  (I don't work in a field where they seem to be common.)  Why would one be concerned about getting a manuscript on arXiv, if it's already under review with a journal?   Is it because of the long publishing timeline of journals?  (That could explain why computer science &mdash; which publishes much more in conferences &mdash; doesn't seem to do much with them.) Complex integral over a circle In an exercise, I'm supposed to assume that $|a| &lt; r &lt; |b|$ and prove that $\displaystyle\int_{\gamma} \frac{1}{(z - a)(z - b)} dz = \frac{2 \pi i}{a - b}$, where $\gamma$ is a circle of radius $r$ centered at the origin with positive orientation. So I had the idea to express this integral as $\frac{1}{a - b} \displaystyle\int_{\gamma} \frac{1}{z - a} - \frac{1}{z - b} dz$. Then I tried to evaluate each of these separately, but I don't really know what to do. I get $\displaystyle\int_{0}^{2 \pi} \frac{i r e^{it}}{r e^{it} - a} dt$ for the first term. How am I supposed to integrate this? I can't substitute $u = re^{it}$, can I? If I select a feature and run self.iface.actionIdentify().trigger() it will open the identify window but won't open the selected feature's data, how can I make it open the data of the selected feature? How do I show that the equation $x^3+10x^2-100x+1729$$=0$ has at least one complex root $a$ such that $|a|$$&gt;$$12$. Check if user is using IE I've been doing some practice questions in the textbook for an upcoming predicate calculus lecture and I think I've managed to get A and B (possibly C) correct but I am clueless on how I can manage D and E.   For A I got: ∃x(R(x)∧¬B)  For B I got: ∀x(¬R(x)∧B)  For C I got: ∃x(R(x)∧(x=y))   - Not sure if this is even remotely correct    R(X) : ”X is rich”  X = Y : ”X equals Y”  B : ”Bill Gates”  a) Someone besides Bill Gates is rich.  b) Bill Gates alone is rich.  c) At least two people are rich.  d) Exactly one person is rich.  e) Exactly two persons are rich.  The universe of discourse is the set of all people. Can I mix different RAM modules in PC? I have 4 RAM slots on the motherboard P5K. At the moment I have two 1GB RAM modules and two 2GB RAM modules. Everything is working fine.   Can I buy one 4GB module and put it instead of the 1GB module?  Another question, can I mix MHz values, like that I have two 667MHz modules and two 1066MHz modules? Is this dangerous for the comp? What is the definition of a timelike and spacelike singularity? I'm trying to determine if 1317 and 116 can be written as the sum of three squares?  I have the condition that if it is not of the form $4^{\alpha}(8k+7)$ then it can be written as a sum of three squares, but how do I use this condition to work it out?  Thank you Should I delete my own questions with answers? I don't know much about this,  But I've some of the questions on which some user has answered (not helpful answer) also but I want to delete those questions, but whenever I go to delete that question. It is prompting with     What does it mean? Can someone tell me what would be the action taken by Stack Exchange if I delete more questions? A subring of the field of fractions of a PID is a PID as well. Let $A$ be a PID and $R$ a ring such that $A\subset R \subset \operatorname{Frac}(A)$, where $\operatorname{Frac}(A)$ denotes the field of fractions of $A$. How to show $R$ is also a PID?   Any hints? Does constant charging harm my Android cellphone? I usually disconnect my cellphone from the charger some time after it's fully charged because I'm afraid the battery could lose capacity over time if I don't. However, I believe modern electronics should be able to handle this kind of scenario and automatically stop charging the battery once it's full. Does this apply to modern Android phones? Or do I need to continue disconnecting it all the time? I have to do some manual reordering or rows in Excel and cut/paste is too hard to use. I'm looking for something that would enable me to use drag'n'drop or to add some buttons to move-up/down move-top/bottom.