Suppose that $f'$ exists and is continuous on $[a, b]$ and $f''$ exists on $(a, b)$, where $a &lt; b$. Show that $\exists\ c \in (a, b)$ such that $$f(b) = f(a) + (b - a)f'(a) + \frac {(b - a)^2} {2} f''(c)\ .$$ From experience (though it is very little), my first go at this question was to use the Mean Value Theorem on $f'$, which allowed me to arrive at $$f''(c) = \frac {f'(b) - f'(a)} {b - a}\ .$$ However, I am not sure where to go from here. I also tried to &quot;work backwards&quot;, by playing around with the given equation, but to no avail. I believe I need to derive some information about $f$, since $f(a)$ and $f(b)$ are in play, but I am not sure what information I can obtain about $f$? A hunch tells me I need to use the mean value theorem on $f$ once more, but I do not know how to proceed. Any hints or intuitions would be greatly appreciated! Edit The calculus module I am taking in university is only an introductory one and does not cover Taylor's Theorem, so I believe my professor is looking for another neat way to approach this problem, perhaps simply by using the mean value theorem. Also, this problem appears in the &quot;higher-difficulty&quot; part of the tutorial-in-question, so I suppose the answer is not so straightforward.
I started reading "The Mathematics of nonlinear Programming - AL peressini" recently.  Theorem : Suppose that $f(x) , f'(x) ,f''(x)$ exist on the closed interval $[a,b]$. If $x^* ,x $ are two different points of $[a,b]$ , then there exists a point $z$ strictly between $x^*$ and $x$ such that   $$f(x)=f(x^*)+f'(x^*)(x-x^*)+\frac{f''(z)}{2}(x-x^*)^2$$  I tried to prove this theorem using The Fundamental theorem of calculus. But I can not stop it at the second derivative . Do I have to use mean value theorem ? or is there a simple proof for this ?
Do I need a UK visa? How and where to apply? How to extend my root (/) partition? Below is the text in the file:  Pseudo name=Apple Code=42B state=fault  Pseudo name=Prance Code=43B state=good   I need to grep for "42B" and get the output from the above text like:   Pseudo name=Apple Code=42B state=fault   Does anyone have idea on how to achieve this using grep/awk/sed? Prove $\csc(x)=\sum_{k=-\infty}^{\infty}\frac{(-1)^k}{x+k\pi}$ Concurrent updates to GDAL (to 2.1.3) and QGIS (to 2.18.5) on Linux Mint 18.1 from the Ubuntugis repos breaks QGIS, with a noted dependency for the virtual package gdal-abi-2-1-2 being uninstallable.    The stable Ubuntugis ppa currently supplies GDAL 2.1.0 (gdal-abi-2-1-0) while the Ubuntugis-unstable ppa has been updated to GDAL 2.1.3 (gdal-abi-2-1-3).  Has anyone found a way around this?    Update:  QGIS bug report for those interested is here:     UPDATE 2  The ticket has been closed as fixed. Everything should be back to normal now. How do you get a submit button in the Windows Phone keyboard? 4 cards are drawn from a pack without replacement. What is the probability of getting all 4 from different suits?  Here's how I tried to solve:  For the first draw, we have 52 cards, and we have to pick one suit. So, probability for this is $\frac{13}{52}$.  For the second draw, only 51 cards are left. The second suit has to be selected, so there are 13 cards from that suit. The probability is $\frac{13}{51}$.  Similarly, the third and fourth draw have probabilities $\frac{13}{50}$ and $\frac{13}{49}$ respectively.  Since the draws are independent, the total probability becomes $$\frac{13}{52} \times \frac{13}{51} \times \frac{13}{50} \times \frac{13}{49}$$  But my book says the answer is $\frac{{13\choose 1} \times {13 \choose 1} \times {13\choose1} \times {13\choose1}}{52 \choose 4}$.  My answer differs by a factor of $4!$. What did I do wrong? Why does Ubuntu 12.10 not detect my Win7 partition? Audits bug in the filtered review queue Just recently the . I've got  this morning which seems to be out of scope for my filtered list. I have filtered my list to only review VBA and Excel-VBA tags while the audit is tagged with perl http pipes which have nothing in common with VBA. I see a problem with the way the system currently works  Since I have a filtered list of tags, once I see a differently tagged question I already know it's an audit. Come on, that's too easy!  Can this please be fixed? update I have ran into  which I decided to edit - the question was tagged VBA and I thought it had nothing to do with VBA so I selected Edit which was the correct action. Now, once the first audit was clearly one to close - nothing to do with VBA and really easy to know that it actually was an audit - the second one was a very good audit. I am just thinking now if this is a proper bug or just a matter of malfunction in the system. I think the first one may have been made an audit for the wrong reason. Let's turn this into a discussion as I can't clearly work the system out. What are the powers shown by the Rings of Power? How to search for only folders in Windows 7 instead of folders AND files Segmentation fault on large array sizes We know that $12^2 = 144$ and that $38^2 = 1444$. Are there any other perfect squares in the form of $\frac{13}{9} (10^n - 1) + 1$ (i.e. $1$ followed by $n$ $4$'s), and how would we prove it? How can I change a cell's background color based on the cell's current value? I have a column with function values of True or False:  =IF(ISNA(A1), false, true)   from row 1 to 5000+ rows. I would like to mark all the False as red background color or text as red. Is there any function available? Why is "what can I ask about here" not the first entry in the help center? On a lot of sites, the first section of the Help Center looks like this:    When I go to a site's help center it's almost always to find out what's on-topic there, because I'm considering asking or migrating a question.  Leading with what not to ask seems odd to me, and I suspect that to someone not already familiar with SE it could be a little off-putting.  It's not always like this, though; here on MSE we get this, which I've seen on some other sites too:    That ordering makes a lot more sense to me.  It's also consistent with the tour.  (Maybe "how do I ask a good question?" should also be "above the fold", in the top five somewhere.)  Why the difference?  Can we either make the order consistent across the network and more logical, or give users or moderators the ability to change the order on their sites? If every real-valued continuous function is bounded on $X$ (metric space), then $X$ is compact. Let $X$ be a metric space. Prove that if every continuous function $f: X \rightarrow \mathbb{R}$ is bounded, then $X$ is compact.  This has been asked before, but all the answers I have seen prove the contrapositive. Realistically, this may be the way to go, but is there way to exhibit an unbounded continuous function (under the assumption $X$ is not compact) without appealing to results beyond introductory real analysis (e.g., the solution I've seen involves the Tietze Extension theorem)?  Because we're working with metric spaces, it's clear that the assumption that $X$ is noncompact (towards proving contrapositive) will lead us to extract a sequence of points in the space with no convergent subsequences. But how can we infer the existence of an unbounded continuous on the space knowing only about some sequence of points in this space (without using anything too advanced)? What is the command to open the shutdown dialog? I would like to use a gesture to shutdown/restart my computer and I need to know the command that launches the shutdown dialog: I need to keep a session alive for 30 minutes and then destroy it. algebra - matrices and polynoms $T:\mathbb P_2[x]↦\mathbb P_2[x]$ s.t.:   $T(1)=3+2x+4x^2$, $T(x)=2+2x^2$, $T(x^2)=4+2x+3x^2$.   Is there base $B$ of $\mathbb P_2[x]$ that $[T]_B$= \begin{bmatrix}     2 &amp; 0 &amp; 0\\     -1 &amp; 1 &amp; 0\\     5 &amp; 0 &amp; 4\\  \end{bmatrix}  Please help. Thanks. How were the solar masses and distance of the GW150914 merger event calculated from the signal?