1997 Paper 3 Q2

Year: 1997
Paper: 3
Question Number: 2

Course: LFM Pure and Mechanics
Section: Exponentials and Logarithms

Difficulty: 1700.0 Banger: 1516.3

exponentials and logarithms curve sketching stationary points ln t / t maximum value implicit equations number of solutions

Problem

Let \[\mathrm{f}(t)=\frac{\ln t}t\quad\text{ for }t>0.\] Sketch the graph of \(\mathrm{f}(t)\) and find its maximum value. How many positive values of \(t\) correspond to a given value of \(\mathrm f(t)\)? Find how many positive values of \(y\) satisfy \(x^y=y^x\) for a given positive value of \(x\). Sketch the set of points \((x,y)\) which satisfy \(x^y=y^x\) with \(x,y>0\).

No solution available for this problem.

Rating Information

Difficulty Rating: 1700.0

Difficulty Comparisons: 0

Banger Rating: 1516.3

Banger Comparisons: 3

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Problem source

Let \[\mathrm{f}(t)=\frac{\ln t}t\quad\text{ for }t>0.\]
Sketch the graph of $\mathrm{f}(t)$ and find its maximum
value. How many positive values of $t$ correspond to a
given value of $\mathrm f(t)$?
Find how many positive values of $y$ satisfy
\(x^y=y^x\) for a given positive value of $x$. Sketch the
set of points $(x,y)$ which satisfy \(x^y=y^x\) with $x,y>0$.

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