| course |
course_year |
question_number |
tags |
title |
year |
Variational Principles |
IB |
78 |
IB |
2019 |
Variational Principles |
|
Paper 3, Section I, A |
2019 |
The function $f$ with domain $x>0$ is defined by $f(x)=\frac{1}{a} x^{a}$, where $a>1$. Verify that $f$ is convex, using an appropriate sufficient condition.
Determine the Legendre transform $f^{}$ of $f$, specifying clearly its domain of definition, and find $\left(f^{}\right)^{*}$.
Show that
$$\frac{x^{r}}{r}+\frac{y^{s}}{s} \geqslant x y$$
where $x, y>0$ and $r$ and $s$ are positive real numbers such that $\frac{1}{r}+\frac{1}{s}=1$.