2015-78.md

course	course_year	question_number	tags	title	year
Variational Principles	IB	78	IB 2015 Variational Principles	Paper 3, Section $I$, A	2015

(a) Define what it means for a function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ to be convex.

(b) Define the Legendre transform $f^{}(p)$ of a convex function $f(x)$, where $x \in \mathbb{R}$. Show that $f^{}(p)$ is a convex function.

(c) Find the Legendre transform $f^{}(p)$ of the function $f(x)=e^{x}$, and the domain of $f^{}$.