Let $x$ be a real number such that $x&gt; 0$.   Prove that $x + \frac{1}{4x} \geq 1$.  I tried to do this but cannot prove it.
Let $x$ be a real number such that $x &gt; 0$. Prove that x +$\frac {1} {4x}  ≥ 1$.   Not really sure on the correct way to approach it/is valid and could use some help.  Answer:  Proof Strategy: Proof by cases:   $x = 1$ $x &gt; 1$ $x &lt; 1$   --  Case 1: $x = 1$  $x+\frac1{4x}$  $= (1) + 1/(4(1))$  $= 1.25 \ge 1$  Case is true  Case 2: $x &gt; 1$, in this case $x = 2$.  $x+\frac1{4x} $  $= (2) + 1/(4(2))$  $= 2.125 \ge 1$  Case is true  Case 3: $x &lt; 1$, in this case $x = 0.5$.  $= (0.5) + 1/(4(0.5))$  $= 1  \ge 1$  Case is true  Since all possible cases were satisfied therefore $x+\frac1{4x} \ge 1 $ when $x &gt; 0$.
This question attempts to collect the few pearls among the dozens of bad C++ books that are published every year. Unlike many other programming languages, which are often picked up on the go from tutorials found on the Internet, few are able to quickly pick up C++ without studying a well-written C++ book. It is way too big and complex for doing this. In fact, it is so big and complex, that there are very many very bad C++ books out there. And we are not talking about bad style, but things like sporting glaringly obvious factual errors and promoting abysmally bad programming styles. Please edit the accepted answer to provide quality books and an approximate skill level — preferably after discussing your addition in . (The regulars might mercilessly undo your work if they disagree with a recommendation.) Add a short blurb/description about each book that you have personally read/benefited from. Feel free to debate quality, headings, etc. Books that meet the criteria will be added to the list.  Books that have reviews by the Association of C and C++ Users (ACCU) have links to the review. *Note: FAQs and other resources can be found in the  and under . I wanted to tidy up a TeX document and was curious if you can use \lstinputlisting or a similar function to include a certain part of a source-code file.  Example, I have a 500 or so line source file and I want to include just lines 300-400.  Is there a way to do this? I keep running into areas where I see a chest in a section of the room that seems completely walled-off. Is there any way to get to these, or is that just the randomly generated dungeon being a tease?  I had to splice two screenshots together to show an example of what I'm talking about:   Is there any way to get in there? Why is causality preserved in special relativity? PART 1: I was reading the article  Wikipedia. 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From the man page on git-merge, there are a number of merge strategies you can use.     resolve -  This can only resolve two heads (i.e. the current branch and another branch you pulled from) using 3-way merge algorithm. It tries to carefully detect criss-cross merge ambiguities and is considered generally safe and fast. recursive -  This can only resolve two heads using 3-way merge algorithm. When there are more than one common ancestors that can be used for 3-way merge, it creates a merged tree of the common ancestors and uses that as the reference tree for the 3-way merge. This has been reported to result in fewer merge conflicts without causing mis-merges by tests done on actual merge commits taken from Linux 2.6 kernel development history. Additionally this can detect and handle merges involving renames. 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