Year: 2007
Paper: 2
Question Number: 7
Course: LFM Pure and Mechanics
Section: Exponentials and Logarithms
Although the paper was by no means an easy one, it was generally found a more accessible paper than last year's, with most questions clearly offering candidates an attackable starting-point. The candidature represented the usual range of mathematical talents, with a pleasingly high number of truly outstanding students; many more who were able to demonstrate a thorough grasp of the material in at least three questions; and the few whose three-hour long experience was unlikely to have been a particularly pleasant one. However, even for these candidates, many were able to make some progress on at least two of the questions chosen. Really able candidates generally produced solid attempts at five or six questions, and quite a few produced outstanding efforts at up to eight questions. In general, it would be best if centres persuaded candidates not to spend valuable time needlessly in this way – it is a practice that is not to be encouraged, as it uses valuable examination time to little or no avail. Weaker brethren were often to be found scratching around at bits and pieces of several questions, with little of substance being produced on more than a couple. It is an important examination skill – now more so than ever, with most candidates now not having to employ such a skill on the modular papers which constitute the bulk of their examination experience – for candidates to spend a few minutes at some stage of the examination deciding upon their optimal selection of questions to attempt. As a rule, question 1 is intended to be accessible to all takers, with question 2 usually similarly constructed. In the event, at least one – and usually both – of these two questions were among candidates' chosen questions. These, along with questions 3 and 6, were by far the most popularly chosen questions to attempt. The majority of candidates only attempted questions in Section A (Pure Maths), and there were relatively few attempts at the Applied Maths questions in Sections B & C, with Mechanics proving the more popular of the two options. It struck me that, generally, the working produced on the scripts this year was rather better set-out, with a greater logical coherence to it, and this certainly helps the markers identify what each candidate thinks they are doing. Sadly, this general remark doesn't apply to the working produced on the Mechanics questions, such as they were. As last year, the presentation was usually appalling, with poorly labelled diagrams, often with forces missing from them altogether, and little or no attempt to state the principles that the candidates were attempting to apply.
Difficulty Rating: 1600.0
Difficulty Comparisons: 0
Banger Rating: 1516.0
Banger Comparisons: 1
A function $\f(x)$ is said to be concave on some interval if $\f''(x)<0$ in that interval. Show that $\sin x$ is concave for $0< x < \pi$ and that $\ln x$ is concave for $x > 0$.
Let $\f(x)$ be concave on a given interval and let $x_1$, $x_2$, $\ldots$, $x_n$ lie in the interval. \textit{Jensen's inequality} states that
\[
\frac1 n \sum_{k=1}^n\f(x_k) \le \f \bigg (\frac1 n
\sum_{k=1}^n x_k\bigg)
\]
and that equality holds if and only if $x_1=x_2= \cdots =x_n$. You may use this result without proving it.
\begin{questionparts}
\item
Given that $A$, $B$ and $C$ are angles of a triangle, show that \[
\sin A + \sin B + \sin C \le \frac{3\sqrt3}2 \,.
\]
\item
By choosing a suitable function $\f$, prove that
\[
\sqrt[n]{t_1t_2\cdots t_n}\; \le \; \frac{t_1+t_2+\cdots+t_n}n
\]
for any positive integer $n$ and for any positive numbers $t_1$, $t_2$, $\ldots$, $t_n$.
Hence:
\begin{enumerate}
\item show that $x^4+y^4+z^4 +16 \ge 8xyz$, where $x$, $y$ and $z$ are any positive numbers;
\item find the minimum value of $x^5+y^5+z^5 -5xyz$,
where $x$, $y$ and $z$ are any
positive numbers.
\end{enumerate}
\end{questionparts}\begin{align*}
&& f(x) &= \sin x \\
\Rightarrow && f''(x) &= -\sin x
\end{align*}
which is clearly negative on $(0,\pi)$ since $\sin$ is positive on this interval.
\begin{align*}
&& f(x) &= \ln x \\
\Rightarrow && f''(x) &= -1/x^2
\end{align*}
which is clearly negative for $x > 0$
\begin{questionparts}
\item Since $A,B,C$ are angles in a triangle, we must have $0 < A,B,C< \pi$ and so we can apply Jensen with $f = \sin$ to obtain:
\begin{align*}
&&\frac13( \sin A + \sin B + \sin C) &\leq \sin \left ( \frac{A+B+C}{3}\right) \\
&&&= \sin \frac{\pi}{3} = \frac{\sqrt{3}}2 \\
\Rightarrow && \sin A + \sin B + \sin C &\leq\frac{3\sqrt{3}}2
\end{align*}
\item Suppose $f(x) = \ln x$, then applying Jensen on the positive numbers $t_1, \ldots, t_n$ we obtain
\begin{align*}
&& \frac1n \left ( \sum_{i=1}^n \ln t_n \right) &\leq \ln \left ( \frac1n\sum_{i=1}^n t_n \right) \\
\Rightarrow && \frac1n \ln\left (\prod_{i=1} t_n\right)&\leq \ln \left ( \frac1n\sum_{i=1}^n t_n \right) \\
\Rightarrow && \ln\left (\left (\prod_{i=1} t_n\right)^{1/n}\right)&\leq \ln \left ( \frac1n\sum_{i=1}^n t_n \right) \\
\Rightarrow && \left (\prod_{i=1} t_n\right)^{1/n}&\leq\frac1n\sum_{i=1}^n t_n \\
\Rightarrow && \sqrt[n]{t_1t_2 \cdots t_n}&\leq\frac1n(t_1 + t_2 + \cdots + t_n) \tag{AM-GM}\\
\end{align*}
\begin{enumerate}
\item Applying AM-GM with $t_1 = x^4, t_2 = y^4, t_3 = z^4, t_4 = 2^4$ we have
\begin{align*}
&& \frac{x^4+y^4+z^4+16}{4} & \geq \sqrt[4]{x^4y^4z^42^4} \\
\Rightarrow && x^4+y^4+z^4+16 &\geq 8xyz
\end{align*}
\item Applying AM-GM with $t_1 = x^5, t_2 = y^5, t_3 = z^5, t_4 = 1^5, t_5 = 1^5$ we have
\begin{align*}
&& \frac{x^5+y^5+z^5+1+1}{5} & \geq \sqrt[5]{x^5y^5z^5} \\
\Rightarrow && x^5+y^5+z^5+2 &\geq 5xyz \\
\Rightarrow && x^5+y^5+z^5 - 5xyz &\geq -2
\end{align*}
Therefore the minimum is $-2$
\end{enumerate}
\end{questionparts}
Not very many candidates attempted this question, but those who did usually found it to be relatively straightforward. It was only the very last part that required much thought, and this was where most attempts lost a few marks. A small number of efforts failed to get beyond part (ii); this was due to not finding a suitable function to work with that gave what turns out to be the Arithmetic Mean-Geometric Mean Inequality. This was a bit of a shame, since the question actually gives the log. function at the very beginning, along with the sine function, which is used in (i).